Non-integrable Ising Models in Cylindrical Geometry: Grassmann Representation and Infinite Volume Limit

Ann. Henri Poincaré 23 (2022), 1061–1139 c 2021 The Author(s)  1424-0637/22/031061-79 published online October 11, 2021 https://doi.org/10.1007/s00023-021-01107-3 Annales Henri Poincaré Non-integrable Ising Models in Cylindrical Geometry: Grassmann Representation and Infinite Volume Limit Giovanni Antinucci, Alessandro Giuliani and Rafael L. Greenblatt Abstract. In this paper, meant as a companion to Antinucci et al. (Energy correlations of non-integrable Ising models: the scaling limit in the cylinder, 2020. arXiv: 1701.05356), we consider a class of non-integrable 2D Ising models in cylindrical domains, and we discuss two key aspects of the multiscale construction of their scaling limit. In particular, we provide a detailed derivation of the Grassmann representation of the model, including a self-contained presentation of the exact solution of the nearest neighbor model in the cylinder. Moreover, we prove precise asymptotic estimates of the fermionic Green’s function in the cylinder, required for the multiscale analysis of the model. We also review the multiscale construction of the effective potentials in the infinite volume limit, in a form suitable for the generalization to finite cylinders. Compared to previous works, we introduce a few important simplifications in the localization procedure and in the iterative bounds on the kernels of the effective potentials, which are crucial for the adaptation of the construction to domains with boundaries. Contents 1. 2. Introduction The Nearest-Neighbor Model 2.1. Diagonalization of the Free Action 2.1.1. Introduction to the Grassmann Variables and Representation 2.1.2. Diagonalization of St1 ,t2 2.2. The Critical Propagator: Multiscale Decomposition and Decay Bounds 1062 1069 1070 1070 1071 1075 1062 G. Antinucci et al. 2.2.1. 2.2.2. Ann. Henri Poincaré Multiscale and Bulk/Edge Decompositions Decay Bounds and Gram Decomposition: Statement of the Main Results 2.3. Asymptotic Behavior of the Critical Propagator 2.4. Symmetries of the Propagator 3. Grassmann Representation of the Generating Function 4. The Renormalized Expansion in the Full-Plane Limit 4.1. Effective Potentials and Kernels: Representation and Equivalence 4.2. Localization and Interpolation 4.3. Trees and Tree Expansions 4.4. Bounds on the Kernels of the Full Plane Effective Potentials 4.5. Beta Function Equation and Choice of the Counterterms Acknowledgements A. Diagonalization of the Matrix Ac B. Proof of Proposition 2.3 C. Proof of Proposition 2.9 D. Non-interacting Correlation Functions in the Scaling Limit References 1075 1076 1078 1079 1080 1086 1089 1091 1096 1102 1110 1115 1116 1120 1129 1135 1137 1. Introduction In this article, which is a companion to [4], we consider a class of non-integrable perturbation of the 2D nearest-neighbor Ising model in cylindrical geometry and discuss some of the key ingredients required in the multiscale construction of the scaling limit of the energy correlations in finite domains. The material presented here generalizes and simplifies the approach proposed by two of the authors in [18], where a similar problem in the translationally invariant setting was investigated. As discussed extensively in [4, Section 3.1], which we refer to for additional motivations and references, the methods of [18], as well as of several other related works on the Renormalization Group (RG) construction of the bulk scaling limit of non-integrable lattice models at the critical point, are insufficient for controlling the effects of the boundaries at the precision required for the construction of the scaling limit in finite domains. This is a serious obstacle in the program of proving conformal invariance of the scaling limit of statistical mechanics models [17]; the goal would be to prove results comparable to the remarkable ones obtained for the nearest neighbor 2D Ising model [10,12,29], but for a class of non-integrable models, such as perturbed Ising [2] or dimer models [22] in two dimensions, via methods that do not rely on the exact solvability of the microscopic model. In this paper and in its companion [4], we attack this program by constructing the scaling limit of the energy correlations of a class of non-integrable perturbations of the standard 2D Ising model in the simplest possible finite domain with boundary, that is, a finite cylinder. Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1063 Let us define the setting more precisely. For positive integers L and M , with L even, we let GΛ be the discrete cylinder with sides L and M in the horizontal and vertical directions, respectively, with periodic boundary conditions in the horizontal direction and open boundary conditions in the vertical directions. We consider GΛ as a graph with vertex set Λ = ZL × (Z ∩ [1, M ]), where ZL = Z/LZ (in the following we shall identify the elements of ZL with {1, . . . , L}, unless otherwise stated) and edge set BΛ consisting of all pairs of the form1 {z, z + êj } for z ∈ Λ, j ∈ {1, 2} and ê1 , ê2 the unit vectors in the two coordinate directions. For x ∈ BΛ , we let j(x) be the j for which x = {z, z + êj } for some z ∈ Λ, so that j(x) = 1 for a horizontal bond and j(x) = 2 for a vertical bond. The model is defined by the Hamiltonian   Jj(x) ǫx − λ V (X)σX , (1.1) HΛ (σ) = − x∈BΛ X⊂Λ where J1 , J2 are two positive constants, representing the couplings in the horizontal and vertical directions, ǫx = ǫx (σ) := σz σz′ for x = {z, z ′ }; the spin Λ variable σ belongs to ΩΛ := {±1} , and σX := x∈X σx ; V is a finite range, translationally invariant, even interaction, obtained by periodizing in the horizontal direction a Λ-independent, translationally invariant, potential on Z2 ; finally, λ is the strength of the interaction, which can be of either sign and, for most of the discussion below, the reader can think of as being small, compared to J1 , J2 , but independent of the system size. In the following, we shall refer to model (1.1) with λ = 0 as to the ‘interacting’ model, in contrast with the standard nearest-neighbor model, which we will refer to as the ‘non-interacting’, one of several terminological conventions motivated by analogy with quantum field theory. The Hamiltonian defines a Gibbs measure ·β,Λ depending on the inverse temperature β > 0, which assigns to any F : ΩΛ → R the expectation value  e−βHΛ (σ) F (σ)  Λ . (1.2) F β,Λ := σ∈Ω −βHΛ (σ) σ∈ΩΛ e The truncated correlations, or cumulants, of the energy observable ǫx , denoted ǫx1 ; · · · ; ǫxn β,Λ , are given by    ∂n log eA1 ǫx1 +···An ǫxn β,Λ  . ǫx1 ; · · · ; ǫxn β,Λ := ∂A1 · · · ∂An A1 =···=An =0 (1.3) For the formulation of the main result, let us fix once and for all an interaction V with the properties spelled out after (1.1), and assume that J1 /J2 and L/M belong to a compact K ⊂ (0, +∞). We let tl := tl (β) := tanh βJl , with l = 1, 2, and recall that in the non-interacting case, λ = 0, the critical temperature βc (J1 , J2 ) is the unique solution of t2 (β) = (1 − t1 (β))/(1 + t1 (β)). Note that there exists a suitable compact K ′ ⊂ (0, 1) such that whenever J1 /J2 ∈ K and β ∈ [ 21 βc (J1 , J2 ), 2βc (J1 , J2 )], then t1 , t2 ∈ K ′ . From now on, we will 1 If z = ((z)1 , (z)2 ) ∈ Λ has horizontal coordinate (z)1 = L, we use the convention that z + ê1 ≡ (1, (z)2 ). 1064 G. Antinucci et al. Ann. Henri Poincaré think K, K ′ to be fixed once and for all. Moreover, we parameterize the Gibbs measure in terms of tl as follows: ·β,Λ ≡ ·λ,t1 ,t2 ;Λ . Given these premises, we are ready to state the main result proven in [4]. Theorem 1.1. Fix V as discussed above. Fix J1 , J2 so that J1 /J2 belongs to the compact K introduced above. There exist λ0 > 0 and analytic functions βc (λ), t∗1 (λ), Z1 (λ), Z2 (λ), defined for |λ| ≤ λ0 , such that, for any finite cylinder Λ with L/M ∈ K and any m-tuple x = (x1 , . . . xm ) of distinct elements of BΛ , with m1 horizontal elements, m2 vertical elements, and m = m1 + m2 ≥ 2,  m2 m1  Z2 (λ) ǫx1 ; . . . ; ǫxm λ,t1 (λ),t2 (λ);Λ = Z1 (λ) ǫx1 ; . . . ; ǫxm 0,t∗1 (λ),t∗2 (λ);Λ +RΛ (x), (1.4) where t1 (λ) := tanh(βc (λ)J1 ), t2 (λ) := tanh(βc (λ)J2 ) and t∗2 (λ) := (1 − t∗1 (λ))/(1 + t∗1 (λ)). Moreover, denoting by δ(x) the tree distance of x, i.e., the cardinality of the smallest connected subset of BΛ containing the elements of x, and by d = d(x) the minimal pairwise distance among the midpoints of the edges in x and the boundary of Λ, for all θ ∈ (0, 1) and ε ∈ (0, 1/2) and a suitable Cθ,ε > 0, the remainder RΛ can be bounded as m |RΛ (x)| ≤ Cθ,ε |λ|m! 1 dm+θ d δ(x) 2−2ε . (1.5) As a corollary of this theorem, one readily obtain the existence and explicit structure of the scaling limit for the ‘energy sector’ of the interacting model, with quantitative estimates on the speed of convergence; see [4, Corollary 1.2] and Appendix D. The proof of Theorem 1.1 is based on a multiscale analysis of the generating function of the energy correlations, formulated in the form of a Grassmann (Berezin) integral. While the strategy of this proof is based on the same general ideas used in [18] in the translationally invariant setting, that is, on the methods of the fermionic constructive RG, the presence of boundaries introduces several technical and conceptual difficulties, whose solution requires to adapt, improve and generalize the ‘standard’ RG procedure (e.g., in the definition of the ‘localization procedure’, in the way in which the kernels of the ‘effective potentials’ are iteratively bounded and in which the resulting bounds are summed over the label specifications, etc.) As discussed in [4, Section 3.1], which we refer to for additional details, we expect that understanding how to implement RG in the presence of boundaries or, more in general, of defects breaking translational invariance, will have an impact on several related problems, such as the computation of boundary critical exponents in models in the Luttinger liquid universality class, the Kondo problem, the Casimir effect, and the phenomenon of many-body localization. In this paper we give a full presentation of some of the key ingredients required in the proof of our main result, namely: Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1065 1. exact solution of the nearest neighbor model on the cylinder in its Grassmann formulation, including multiscale bounds on the bulk and edge parts of the fermionic Green’s function (Sect. 2); 2. reformulation of the generating function of energy correlations of the interacting model as a Grassmann integral (Sect. 3); 3. tree expansion and iterative bounds on the kernels of the effective potentials of the interacting theory in the full plane limit, including the computation and proof of analyticity of the interacting critical temperature (Sect. 4). The other ingredients, including most of the novel aspects of the RG construction in finite volume, such as the definition of the localization procedure in finite volume, the norm bounds on the edge part of the effective potentials and the asymptotically sharp estimates on the correlation functions in the cylinder, are deferred to [4]; see the end of [4, Section 3.1] for a detailed summary and roadmap of the proof of Theorem 1.1. Before starting the technical presentation, let us anticipate in little more detail the contents of the following sections, thus clarifying the main results of this paper. Section 2: exact solution of the model in the cylinder. The multiscale construction of the interacting theory in the domain Λ requires a very fine control of the non-interacting model at the critical point, and, in particular, of the structure of its fermionic Green’s function, which we call the ‘propagator’; the propagator is nothing but the inverse of a signed adjacency matrix A, whose definition we recall in Sect. 2 below [26, Chapter IX]. The key properties we need, and we prove in Sect. 2 below (with some—important!—technical aspects of the proofs deferred to Appendices A, B and C ), see, in particular, Eq. (2.2.14) and Proposition 2.3 below, are the following: • multiscale decomposition of the propagator and bulk–edge decomposition of the single-scale propagator; • exponentially decaying pointwise bounds on the bulk and edge parts of the single-scale propagators, with optimal dimensional bounds (with respect to the scale index) on their L∞ norms and on their decay rates; • Gram representation2 of the bulk and edge parts of the single-scale propagators, with optimal dimensional bounds (with respect to the scale index) on the norms of the Gram vectors. In reference with the second item, let us remark that the exponential decay needed (and proved below) for the propagator between two points z, z ′ ∈ Λ, is in terms of the ‘right’ distance between z and z ′ , namely: the standard Euclidean distance on the cylinder between z and z ′ in the case of the bulk part of the single-scale propagator; the Euclidean distance on the cylinder between z, z ′ and the boundary of Λ, in the case of the edge part of the single-scale propagator. In particular, the exponential decay of the edge part 2 We say that a matrix g admits a Gram representation, if its elements gi,j can be written as the scalar product of two vectors ui and vj in a suitable Hilbert space. 1066 G. Antinucci et al. Ann. Henri Poincaré of the single-scale propagator in the distance of z, z ′ from the boundary of Λ is of crucial importance for proving improved dimensional bounds on the finite-size corrections to the thermodynamic and correlation functions of the interacting model, which are systematically used in the conclusion of the proof of Theorem 1.1 in the companion paper [4], see [4, Section 4]. The proof we give of these key properties is based on an exact diagonalization of the signed adjacency matrix A in terms of the roots of a set of polynomials (this calculation first appeared in [23], and a similar calculation for a rectangle appears in [25]). It is unlikely that such an explicit diagonalization can be obtained in more general domains than the torus, the straight cylinder or the rectangle. Therefore, in order to generalize Theorem 1.1 to more general domains, it would be desirable to prove the properties summarized in the three items above via a more robust method, not based on an explicit diagonalization of A. It remains to be seen whether the methods of discrete holomorphicity, which allowed to prove the convergence of the propagator in general domains to an explicit, conformally covariant, limiting function [10], may allow one to prove the desired properties in general domains. Section 3: Grassmann representation of the generating function. In Sect. 3, we turn our attention to the generating function for the energy correlations      βJj(x) + Ax ǫx + βλ exp V (X)σX . (1.6) ZΛ (A) := σ∈ΩΛ x∈BΛ X⊂Λ that, if computed at a configuration A such that  Ax is equal to Ai for x = xi and zero otherwise, reduces to the combination eA1 ǫx1 +···An ǫxn β,Λ appearing in (1.3), up to an overall multiplicative constant, independent of A. In Proposition 3.1 and Eq.(3.22) (adapting a similar result for the torus in [18]), we show that the correlations without repeated bonds are the same as those obtained by replacing ZΛ (A) with a Grassmann integral of the form  (1) W(A ) ∗ ΞΛ (A) := e Pc∗ (Dφ)Pm (Dξ)eV (φ,ξ,A ) , (1.7) ∗ are Gaussian Grassmann measures associated with the critwhere Pc∗ and Pm ical, non-interacting Ising model at parameters t∗1 , t∗2 := (1 − t∗1 )/(1 + t∗1 ), with t∗1 a free parameter. Moreover, W(A) is a multilinear function of A and V (1) (φ, ξ, A) is a Grassmann polynomial whose coefficients are multilinear functions of A, both of which are defined in terms of explicit, convergent, expansions. As a corollary of Lemma 3.2, we additionally prove that the ‘kernels’ of W(A) and V (1) (φ, ξ, A) (i.e., the coefficients of their expansions in A, φ, ξ, thought of as functions of the positions of the components of A, φ, ξ on the cylinder) can be naturally decomposed into sums of a ‘bulk’ part (equal, essentially, to their infinite plane limit restricted to the cylinder, with the appropriate boundary conditions) plus an ‘edge’ part (their boundary corrections), exponentially decaying in the appropriate distances. In particular, the edge part of the kernels decays exponentially (on the lattice scale) away from Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1067 the boundary, a fact that will play a major role in the control of the boundary corrections to the correlation functions in [4]. Let us remark that, in addition to t∗1 and to the inverse temperature β, the representation (1.7) has another free parameter, Z (entering the definition of V (1) ); while, for the validity of (1.7), these parameters can be chosen arbitrarily in certain intervals, in order for this representation to produce a convergent expansion for the critical energy correlations of the interacting model, uniformly in the system size, we will need to fix t∗1 , β, Z appropriately (a posteriori, they will be fixed uniquely by our construction, see below). Parameters of this kind are known as counterterms in the RG terminology. Section 4: the RG expansion for the effective potentials in the full plane limit. Equation (1.7) is the starting point for a multiscale expansion, which is fully presented in the companion paper [4], see in particular [4, Section 3], but which we summarize here in order to provide the context for Sect. 4, where we carry out an auxiliary expansion for the full plane limit of the kernels of the ‘effective potentials’. Such an auxiliary expansion, among other things, fixes the values of t∗1 , β, Z which are actually used in Theorem 1.1, see Sect. 4.5 below. The goal is to iteratively compute (1.7) in terms of a sequence of effective potentials, defined as follows: at the first step we let  (1) W (0) (A )+V (0) (φ,A ) ∗ ∝ Pm (Dξ)eV (φ,ξ,A ) , (1.8) e where ∝ means ‘up to a multiplicative constant independent of A’; the polynomials W (0) , V (0) are specified uniquely by the normalization W (0) (0) = V (0) (0, A) = 0. (0) We are left with computing the integral of eV (φ,A ) with respect to the Gaussian integration Pc∗ (Dφ) with propagator g∗c . As anticipated above, in 0 Sect. 2.2 we decompose the critical propagator g∗c as g(≤h) + j=h+1 g(j) , for any h < 0; correspondingly, in light of the addition formula for Grassmann integrals (see, e.g., [21, Proposition 1]), we introduce the sequences P (≤h) and P (h) of Gaussian Grassmann integrations, whose propagators are g(≤h) and g(h) , respectively, and satisfy, for any Grassmann function f ,   P (≤h) (Dφ)f (φ) = P (≤h−1) (Dφ)P (h) (Dϕ)f (φ + ϕ). (1.9) We can then iteratively define V (h) and W (h) with W (h) (0) = V (h) (0, A) ≡ 0 and  (h−1) (h) (A )+V (h−1) (φ,A ) eW ∝ P (h) (Dϕ)eV (φ+ϕ,A ) . (1.10) The iteration continues until the scale h∗ = −⌊ log2 (min{L, M })⌋ is reached, at which point we let  (h∗ −1) ∗ (h∗ ) (A ) ∝ P (≤h ) (Dφ)eV (φ,A ) , (1.11) eW 1068 G. Antinucci et al. Ann. Henri Poincaré giving ΞΛ (A) ∝ exp W(A) + 0  h=h∗ −1 W (h)  (A) . (1.12) In order to obtain bounds on the kernels of W (h) (A) leading to an expansion for the energy correlations that is uniform in the system size, at each step it is necessary to isolate from V (h) the contributions that tend to expand (in an appropriate norm) under iterations: these, in the RG terminology, are the relevant and marginal terms, which we collect in the so-called local part of V (h) , denoted by LV (h) . In other words, at each step of the iteration, we rewrite V (h) = LV (h) + RV (h) , where, in our case, LV (h) includes: three terms that are quadratic in the Grassmann variables and independent of A, depending on a sequence of h-dependent parameters which we denote υ = {(νh , ζh , ηh )}h≤1 and call the running coupling constants; and two terms that are quadratic in the Grassmann variables and linear in A, depending on another sequence of effective parameters, {Z1,h , Z2,h }h≤0 , called the effective vertex renormalizations. Moreover, RV (h) is the so-called irrelevant, or renormalized, part of the effective potential, which is not the source of any divergence. Such a decomposition corresponds to a systematic reorganization, or ‘resummation’, of the expansions arising from the multiscale computation of the generating function. The goal will be to show that, by appropriately choosing the parameters t∗1 , β, Z, which the right side of (1.7) depends on (and which are related via a simple invertible mapping to the initial values of the running coupling constants, ν0 , ζ0 , η0 ), the whole sequence υ remains bounded, uniformly in h∗ ; see Sect. 4.5. Under these conditions, we will be able to show that the resulting expansions for multipoint energy correlations are convergent, uniformly in h∗ . Our estimates are based on writing the quantities involved as sums over terms indexed by Gallavotti-Nicolò (GN) trees [13–15], which emerge naturally from the multiscale procedure; the relevant aspects of the definitions of the GN trees will be reviewed in Sect. 4.3 below. In order to obtain L, M independent values of these parameters, we study the iteration in the limit L, M → ∞ in Sect. 4; we can also restrict to A = 0, since this already includes all of the potentially divergent terms. This would superficially appear to involve a number of complications such as defining an infinite-dimensional Grassmann integral, but in fact the multiscale computation of the generating function, when understood as an iteration for the kernels (h) of V (h) , denoted by VΛ , has a perfectly straightforward infinite-volume version, which is stated and analyzed in Sect. 4. The convergence as L, M → ∞ (h) (h) of the finite volume kernels VΛ to the solution V∞ of the infinite-volume recursive equations for the kernels is one of the main subjects of [4], especially [4, Section 3]. Section 4 is a reformulation of [18, Section 3]. We nonetheless present it at length, partly because the treatment of the propagator on the cylinder in Sect. 2 imposes a different choice of variables which makes the translation of some statements awkward, but mainly in order to take the opportunity to Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1069 make a number of technical improvements and simplify some unnecessarily obscure aspects of what is already a complicated argument. Previously, e.g., in [3,15,16,18], the localization operator (and consequently the remainder) was defined in terms of the Fourier transform of the functions involved. This has the advantage of providing a simple procedure for parametrizing the local part of the effective potential by a finite number of running coupling constants, but is quite difficult to apply to non-translationinvariant systems (in [3] this led to a peculiar restriction on the dependence of the interaction on the system size). Moreover, it makes the treatment of finite size corrections awkward and leads to a convoluted definition of the derivative operators in the remainder (see [6] and [19] for the treatment of finitesize corrections on a finite torus via the ‘standard’ definition of localization operator). To deal with this, in Sect. 4.2 we introduce a localization operator defined directly in terms of lattice functions and write the remainder in terms of discrete derivatives using a lattice interpolation procedure. Such definitions naturally admit finite volume counterparts, discussed in [4, Section 3.1]. The strategy used to estimate the interpolation factors in the above cited works also involves decomposing them into components which can be matched with propagators; this involves a number of complications, since it cannot be done in a strictly iterative fashion (this is the problem discussed in [6, Section 3.3]). When we handle this issue in Sect. 4.4, see in particular Proposition 4.6, we instead show iteratively that the coefficient function of the effective interaction satisfies a norm bound morally equivalent to exponential decay in the position variables (associated with exponential decay in the scaledecomposed propagators, see Proposition 2.3); this then makes it possible to bound the contribution of the interpolation operator immediately, avoiding technical issues such as the ‘accumulation of derivatives’ (see [6, Section 3.3] and [15, end of Section 8.4]) or the proof that the Jacobian associated with the change of variables arising from the interpolation procedure is equal to 1 (see [6, (3.119)]). In these aspects, the strategy used in this paper for iteratively estimating the kernels of the effective potentials overlaps with [20], which was developed in parallel with the present work. 2. The Nearest-Neighbor Model In this section we review some aspects of the exact solution of the nearestneighbor model (λ = 0), which will play a central role in the multiscale computation of the generating function for the energy correlations of the nonintegrable model, to be discussed in the following sections. In particular, after having recalled the Grassmann representation of the partition function, we explain how to diagonalize the Grassmann action; next, we compute the Grassmann propagator and define its multiscale decomposition, to be used in the following; finally, we compute the scaling limit of the propagator, with quantitative bounds on the remainder. 1070 G. Antinucci et al. Ann. Henri Poincaré 2.1. Diagonalization of the Free Action 2.1.1. Introduction to the Grassmann Variables and Representation. Let us recall the form of the Hamiltonian HΛ (σ) (1.1) in the integrable case λ = 0: HΛ (σ) = − 2  Jl l=1  σz σz+êl , z∈Λ with the understanding that σz+ê2 = 0 for z = (z1 , z2 ) such that3 z2 = M and σz+ê1 = σz+(1−L)ê1 for z such that z1 = L. As is well known [26, Chapter VI.3], the partition function at inverse  −βHΛ (σ) temperature β > 0, ZΛ = e , can be written as a Pfaffian, σ∈ΩΛ which admits the following representation in terms of Grassmann variables, see, e.g., [27] or [16, Appendix A1]:  (2.1.1) ZΛ =2LM (cosh βJ)L(2M −1) DΦ eSt1 ,t2 (Φ) , where Φ = {(H z , Hz , V z , Vz )}z∈Λ is a collection of 4LM Grassmann variables (we will also use the notation {Φi }i∈I for I a suitable label set with 4LM elements), DΦ denotes the Grassmann ‘differential’,  DΦ = dH z dHz dV z dVz , z∈Λ and St1 ,t2 (Φ) :=  (t1 H z Hz+ê1 + t2 V z Vz+ê2 + H z Hz + V z Vz + V z H z z∈Λ +Vz H z + Hz V z + Vz Hz ) (2.1.2) where tl = tanh βJl for l = 1, 2, and H(L+1,(z)2 ) , V((z)1 ,M +1) should be interpreted as representing −H(1,(z)2 ) and 0, respectively. The identification of H(L+1,(z)2 ) with −H(1,(z)2 ) corresponds to anti-periodic boundary conditions in the horizontal direction for the Grassmann variables: these are the right boundary conditions to consider for a cylinder with an even number of sites in the periodic direction, see [26, Eq.(2.6d)]. For later reference, we let Ex = H z Hz+ê1 for a horizontal edge x with endpoints z, z + ê1 , and Ex = V z Vz+ê2 for a vertical edge x with endpoints z, z + ê2 . Sometimes, we will call {(H z , Hz )}z∈Λ the horizontal variables, and {(V z , Vz )}z∈Λ the vertical ones. The quadratic form St1 ,t2 (Φ) can be written as St1 ,t2 (Φ) = 21 (Φ, AΦ) for a suitable 4LM × 4LM anti-symmetric matrix A (here (·, ·) indicates the standard scalar product  for vectors whose components are labeled by indices in I, i.e., (Φ, AΦ) = i,j∈I Φi Aij Φj ). In terms of this matrix A, (2.1.1) can be rewritten as ZΛ = 2LM (cosh βJ1 ) 3 In LM (cosh βJ2 ) L(M −1) PfA. this section we shall denote the components of z ∈ Λ by z1 , z2 . We warn the reader that in the following sections the symbols z1 , z2 will mostly be used, instead, for the first two elements of an n-ple z in Λn or in (Z2 )n , for which we will use the notation z = (z1 , . . . , zn ). Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1071 [We recall that the Pfaffian of a 2n × 2n antisymmetric matrix A is defined as 1  PfA := n (−1)π Aπ(1),π(2) ...Aπ(2n−1),π(2n) ; (2.1.3) 2 n! π where the sum is over permutations π of (1, . . . , 2n), with (−1)π denoting the signature. One of the properties of the Pfaffian is that (PfA)2 = detA.] For later purpose, we also need to compute the averages of arbitrary monomials in the Grassmann variables Φi , with i ∈ I. These can all be reduced to the computation of the inverse of A, thanks to the ‘fermionic Wick rule’:  1 1 (2.1.4) DΦ Φi1 · · · Φim e 2 (Φ,AΦ) = PfG , Φi1 · · · Φim  := PfA where, if m is even, G is the m × m matrix with entries Gjk = Φij Φik  = −[A−1 ]ij ,ik (2.1.5) (if m is odd, the r.h.s. of (2.1.4) should be interpreted as 0). Often Φi Φj  is referred to as the (ij component of the) propagator of the Grassmann field Φ, or as the covariance of DΦeS(Φ) ; such a form (with a quadratic function in the exponent) is known as a Grassmann Gaussian measure. In the following sections, we shall explain how to compute the Pfaffian of A and its inverse A−1 , via a block diagonalization procedure. 2.1.2. Diagonalization of St 1 ,t 2 . Horizontal direction diagonalization and Schur reduction. By exploiting the periodic boundary conditions in the horizontal direction, we can block diagonalize the Grassmann action by performing a Fourier transform in the same direction: for each z2 ∈ {1, 2, . . . , M } we define Hz2 (k1 ) = L  ik1 z1 e H(z1 ,z2 ) , H z2 (k1 ) = z1 =1 Vz2 (k1 ) = L  DL := in terms of which  1 St1 ,t2 (Φ) = L eik1 z1 H (z1 ,z2 ) , z1 =1 ik1 z1 e z1 =1 with k1 ∈ DL , where L  V(z1 ,z2 ) , V z2 (k1 ) = L  (2.1.6) ik1 z1 e V (z1 ,z2 ) , z1 =1  π(2m − 1) L L : m = − + 1, · · · , L 2 2 k1 ∈DL z2 =1,...,M  , (2.1.7)  (1 + t1 e−ik1 )H z2 (−k1 )Hz2 (k1 ) + V z2 (−k1 )Vz2 (k1 ) +t2 V z2 (−k1 )Vz2 +1 (k1 ) + V z2 (−k1 )H z2 (k1 ) + Vz2 (−k1 )H z2 (k1 )  +Hz2 (−k1 )V z2 (k1 ) + Vz2 (−k1 )Hz2 (k1 ) . (2.1.8) [Note that as a consequence of the convention that V(z1 ,M +1) = 0, VM +1 (k1 ) should also be interpreted as 0.] The terms in the second line of (2.1.8), mixing 1072 G. Antinucci et al. Ann. Henri Poincaré the horizontal with the vertical variables, can be eliminated by a linear transformation corresponding to a Schur reduction of the coefficient matrix (cf. [26, p. 120]):        ξ (1 + t1 eik1 )−1 −(1 + t1 eik1 )−1 φ+,z2 (k1 ) (k ) H z2 (k1 ) = +,z2 1 + , ξ−,z2 (k1 ) (1 + t1 e−ik1 )−1 (1 + t1 e−ik1 )−1 φ−,z2 (k1 ) Hz2 (k1 )     φ (k ) V z2 (k1 ) = +,z2 1 ; φ−,z2 (k1 ) Vz2 (k1 ) (2.1.9) Defining a related set of Grassmann variables on Λ by φω,z =  1 e−ik1 z1 φω,z2 (k1 ) and analogously for ξω,z , we then obtain k ∈D L 1 L        L  ξ+,z s+ (z1 − y) −s+ (z1 − y) φ+,(y,z2 ) Hz , = + φ−,(y,z2 ) s− (z1 − y) s− (z1 − y) ξ−,z Hz y=1     φ+,z Vz = , (2.1.10) φ−,z Vz  e−ik1 z1 where s± (z1 ) := L1 k1 ∈DL 1+t ±ik1 . By the Poisson summation formula, 1e  π  dk1 e−ik1 z1 . (−1)n s∞,± (z1 + nL), with s∞,± (z) = s± (z1 ) = ±ik1 −π 2π 1 + t1 e n∈Z (2.1.11) It is straightforward to check, via a complex shift of the path of integration over k1 , that s∞,± (and, therefore, s± ) decays exponentially in z1 ; more precisely, |s∞,± (z1 )| ≤ e−α|z1 | (1 − t1 eα )−1 for any α ∈ [0, − log t1 ), and |s± (z1 ) − s∞,± (z1 )| ≤ e−αL (1 − t1 eα )−1 whenever |z1 | ≤ L/2. In terms of the new variables, the Grassmann action reads St1 ,t2 (Φ) = Sm (ξ) + Sc (φ) (the labels ‘m’ and ‘c’ stand for ‘massive’ and ‘critical’, for reasons that will become clear soon), where Sm (ξ) = M 1   (1 + t1 e−ik1 )ξ+,z2 (−k1 )ξ−,z2 (k1 ), L z =1 k1 ∈DL (2.1.12) 2 M 1   − b(k1 )φ+,z2 (−k1 )φ−,z2 (k1 )+t2 φ+,z2 (−k1 )φ−,z2 +1 (k1 ) Sc (φ) = L k1 ∈DL z2 =1  i i − ∆(k1 )φ+,z2 (−k1 )φ+,z2 (k1 ) + ∆(k1 )φ−,z2 (−k1 )φ−,z2 (k1 ) , 2 2 (2.1.13) with ∆(k1 ) := 2t1 sin k1 , |1 + t1 eik1 |2 b(k1 ) := 1 − t21 , |1 + t1 eik1 |2 (2.1.14) where, as a consequence of the convention used above for Vz , the term in Sc (φ) involving φM +1,− (k1 ) should be interpreted as being equal to zero. Since Sm and Sc involve independent sets of Grassmann variables, the Gaussian integral Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1073 appearing in the partition function factors into a product of two integrals, and the propagators associated with the two terms can be calculated separately. The ‘massive’ propagator The calculations for Sm are trivial. Let the antisymmetric matrix Am be defined by Sm (ξ) = 21 (ξ, Am ξ). Recall that Sm was defined in (2.1.12), from which  PfAm = (1 + t1 eik1 )M , k1 ∈DL and the propagator is given by the appropriate entry of A−1 m , which in the form used in (2.1.12) is block-diagonal with 2 × 2 blocks such that ω ξω,z2 (k1 )ξω′ ,z2′ (k1′ ) = Lδz2 ,z2′ δω,−ω′ δk1 ,−k1′ . 1 + t1 eiωk1 Therefore, going back to x-space, ξω,z ξω′ ,z′  = ω δω,−ω′ sω (z1 − z1′ ) δz2 ,z2′ , (2.1.15) where sω (z1 ) was defined right after (2.1.10). For later reference, the matrix formed by the elements in (2.1.15) will be referred to as the massive propagator and denoted by   0 s+ (z1 − z1′ ) ′ . (2.1.16) gm (z, z ) = δz2 ,z2′ −s− (z1 − z1′ ) 0 Recalling that s± decays exponentially, see comments after (2.1.11), we see that gm (z, z ′ ) decays exponentially as well, and so corresponds to a massive field in the language of quantum field theory. The ‘critical’ propagator The antisymmetric matrix Ac defined by Sc (φ) = 1 2 (φ, Ac φ) can be placed into an explicit block-diagonal form by an ansatz resembling a Fourier sine transform with shifted frequencies; this involves a lengthy but elementary calculation which is detailed in Appendix A. Here we simply state the result for the critical case 1 − t1 1 − t2 t 1 t2 + t 1 + t 2 = 1 ⇔ t2 = ⇔ t1 = , (2.1.17) 1 + t1 1 + t2 which is the only case of relevance for the present work4 . In this case we obtain     g (z, z ′ ) g+− (z, z ′ ) φ+,z φ+,z′  φ+,z φ−,z′  = gc (z, z ′ ) (2.1.18) = ++ g−+ (z, z ′ ) g−− (z, z ′ ) φ−,z φ+,z′  φ−,z φ−,z′  where 1  L k ∈D  ′ 1 e−ik1 (z1 −z1 ) 2NM (k1 , k2 ) 1 L k2 ∈QM (k1 )   −ik2 (z2 +z2′ ) ĝ++ (k1 , k2 ) −ik2 (z2 −z2′ ) × e ĝ(k1 , k2 ) − e ĝ−+ (k1 , k2 ) gc (z, z ′ ) := 4 The  ĝ+− (k1 , −k2 ) e2ik2 (M +1) ĝ−− (k1 , k2 ) (2.1.19) case t2 < (1 − t1 )/(1 + t1 ), corresponding to the paramagnetic phase can be considered in the same way, while the case t2 > (1 − t1 )/(1 + t1 ) is somewhat more complicated since one of the frequencies may be complex [23]. 1074 G. Antinucci et al. with Ann. Henri Poincaré   ĝ++ (k1 , k2 ) ĝ+− (k1 , k2 ) ĝ−+ (k1 , k2 ) ĝ−− (k1 , k2 )   1 −(1 − t21 )(1 − B(k1 )e−ik2 ) −2it1 sin k1 := +2it1 sin k1 D(k1 , k2 ) (1 − t21 )(1 − B(k1 )eik2 ) ĝ(k1 , k2 ) := (2.1.20) where D(k1 , k2 ) := 2(1 − t2 )2 (1 − cos k1 ) + 2(1 − t1 )2 (1 − cos k2 ), (2.1.21) ik1 2 B(k1 ) := t2 |1 + t1 e | , 1 − t21 (2.1.22) QM (k1 ) is the set of solutions of the following equation, thought of as an equation for k2 at k1 fixed, in the interval (−π, π): sin k2 (M + 1) = B(k1 ) sin k2 M, (2.1.23) and NM (k1 , k2 ) = d dk2 (B(k1 ) sin k2 M − sin k2 (M + 1)) B(k1 ) cos k2 M − cos k2 (M + 1) . (2.1.24) Remark 2.1. From the above formula it is immediately clear that ĝ++ (k1 , k2 ) = ĝ++ (k1 , −k2 ) = −ĝ++ (−k1 , k2 ) = ĝ−− (−k1 , k2 ) (2.1.25) and ĝ+− (k1 , k2 ) = ĝ+− (−k1 , k2 ) = −ĝ−+ (k1 , −k2 ); (2.1.26) furthermore, Eq. (2.1.23) is equivalent to ĝ+− (k1 , k2 ) = −e−2ik2 (M +1) ĝ−+ (k1 , k2 ), (2.1.27) which therefore holds for all q ∈ QM (k1 ). Moreover, NM (k1 , k2 ) = NM (−k1 , k2 ) = NM (k1 , −k2 ). As we will see in Sect. 2.4 below, these relationships are closely related to the symmetries of the Ising model on a cylindrical lattice. Remark 2.2. The definition (2.1.19) can be extended to all z, z ′ ∈ R2 ; then in particular, using the relationships listed in the previous remark, we have g++ ((z1 , 0) , z ′ ) = g++ (z, (z1′ , 0)) = g+− ((z1 , 0) , z ′ ) = g−+ (z, (z1′ , 0)) = 0, (2.1.28) and g+− (z, (z1′ , M + 1)) = g−+ ((z1 , M + 1) , z ′ ) = g−− ((z1 , M + 1) , z ′ ) = g−− (z, (z1′ , M + 1)) = 0, (2.1.29) for all z, z ′ . Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1075 2.2. The Critical Propagator: Multiscale Decomposition and Decay Bounds In this section, we decompose gc (z, z ′ ) into a sum of terms satisfying bounds which are the main inputs of the multiscale expansion. 2.2.1. Multiscale and Bulk/Edge Decompositions. Let fη (k1 , k2 ) := D(k1 , k2 )e−ηD(k1 ,k2 ) , (2.2.1) with D(k1 , k2 ) as in (2.1.21). Note that  ∞ fη (k1 , k2 ) dη = 1 (2.2.2) 0 as long as k1 and k2 are not both integer multiples of 2π (and so whenever k2 ∈ QM (k1 )). Comparing with Eq. (2.1.20), we see immediately that fη (k1 , k2 )ĝ(k1 , k2 ) is an entire function of both k1 and k2 . The reader may find it helpful in what follows to bear in mind that, for large η, fη (k1 , k2 ) is peaked in a region where D(k1 , k2 ) is of the order η −1 , and so k1 , k2 are of order η −1/2 . Thanks to (2.2.2), fη induces the following multiscale decomposition of gc defined in (2.1.19). Let h∗ := −⌊ log2 (min{L, M })⌋; then, for any h∗ ≤ h < 0, 0  g(j) (z, z ′ ), (2.2.3) g[η] (z, z ′ ) dη, (2.2.4) g[η] (z, z ′ ) dη for h∗ < h < 0,  ∞ g[η] (z, z ′ ) dη, g(≤h) (z, z ′ ) := (2.2.5) gc (z, z ′ ) = g(≤h) (z, z ′ ) + j=h+1 where (0) g g(h) (z, z ′ ) :=  ′ (z, z ) :=  1 0 2−2h 2−2h−2 (2.2.6) 2−2h−2 and [η] g ′ (z, z ) := 1 L   1 −ik1 (z1 −z1′ ) e fη (k1 , k2 ) 2NM (k1 , k2 )    ĝ+− (k1 , −k2 ) −ik2 (z2 −z2′ ) −ik2 (z2 +z2′ ) ĝ++ (k1 , k2 ) × e ĝ(k1 , k2 ) − e . 2ik2 (M +1) ĝ−+ (k1 , k2 ) e ĝ−− (k1 , k2 ) k1 ∈DL k2 ∈QM (k1 ) (2.2.7) Note that the single-scale propagators preserve the cancellations at the boundary spelled out in Remark 2.2 above, namely, denoting the components of g(h) (h) by gωω′ , with ω, ω ′ ∈ {±}, in analogy with (2.1.18), (h) (h) (h) (h) g++ ((z1 , 0) , z ′ ) = g++ (z, (z1′ , 0)) = g+− ((z1 , 0) , z ′ ) = g−+ (z, (z1′ , 0)) = 0, (h) (h) (h) g+− (z, (z1′ , M + 1)) = g−+ ((z1 , M + 1) , z ′ ) = g−− ((z1 , M + 1) , z ′ ) (h) = g−− (z, (z1′ , M + 1)) = 0, (2.2.8) 1076 G. Antinucci et al. Ann. Henri Poincaré (≤h) and analogously for gωω′ . Note also that, taking L, M → ∞, the cutoff prop[η] agator gc (z, z ′ ) tends to its infinite-plane counterpart, provided that z, z ′ are chosen ‘well inside the cylinder’; in particular, if zL,M := (L/2, ⌊M/2⌋), then lim g[η] (zL,M + z, zL,M + z ′ )  dk1 dk2 −i[k1 (z1 −z1′ )+k2 (z2 −z2′ )] e fη (k1 , k2 )ĝ(k1 , k2 ) (2π)2 L,M →∞ = (2.2.9) [−π,π]2 [η] (z − z ′ ). =: g∞ For later purposes, we need to decompose the cutoff propagator g[η] into a ‘bulk’ part which is minimally sensitive to the size and shape of the cylinder, plus a remainder which we call the ‘edge’ part. The bulk part is simply chosen [η] to be the restriction of g∞ to the cylinder, with the appropriate (anti-periodic) boundary conditions in the horizontal direction: [η] [η] gB (z, z ′ ) := sL (z1 − z1′ ) g∞ (perL (z1 − z1′ ) , z2 − z2′ ) ∈ {1, . . . , L}, ⎧ ′ ⎪ ⎨+1, |z1 − z1 | < L/2 sL (z1 − z1′ ) := 0, |z1 − z1′ | = L/2 ⎪ ⎩ −1, |z1 − z1′ | > L/2 where, recalling that and (2.2.10) z1 , z1′   z1 1 + perL (z1 ) := z1 − L . L 2 (2.2.11) (2.2.12) The edge part is, by definition, the difference between the full cutoff propagator and its bulk part: [η] [η] gE (z, z ′ ) := g[η] (z, z ′ ) − gB (z, z ′ ). (h) (2.2.13) (h) Using these expressions, we define gB and gE via the analogues of (2.2.4)– (2.2.5), with the subscript c replaced by B and E, respectively. As a consequence, for any h∗ ≤ h < 0, gc (z, z ′ ) = g(≤h) (z, z ′ ) + 0  (j) (j) (gB (z, z ′ ) + gE (z, z ′ )). (2.2.14) j=h+1 As already observed in Remark 2.2, all the functions involved in this identity can be naturally extended to all x, y ∈ R2 (and, therefore, in particular, to all z, z ′ ∈ Z2 ), by interpreting the right side of (2.2.7), etc., as a function on R2 × R2 . 2.2.2. Decay Bounds and Gram Decomposition: Statement of the Main Results. Given the multiscale and bulk–edge decomposition (2.2.14), we now intend to prove suitable decay bounds for the single-scale bulk and edge propagators, as well as to show the existence of an inner product representation (‘Gram representation’) thereof. These will be of crucial importance in the non-perturbative multiscale bounds on the partition and generating functions, Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1077 discussed in the rest of this work, and they are summarized in the following proposition.  2 Given a function f : Z2 → C (or a function f : Λ2 → C extend2 2 able to (Z ) , in the sense explained after (2.2.14)), we let ∂1,j be the discrete derivative in direction j with respect to the first argument, defined by ∂1,j f (z, z ′ ) := f (z + êj , z ′ ) − f (z, z ′ ), with êj the j-th Euclidean basis vector; an analogous definition holds for ∂2,j . Proposition 2.3. There exist constants c, C such that, for any integer h in [h∗ + 1, 0], any r = (r1,1 , r1,2 , r2,1 , r2,2 ) ∈ Z4+ , and any z, z ′ ∈ Λ2 , 1. ∂ r g(h) (z, z ′ ) ≤ C 1+|r |1 r!2(1+|r |1 )h e−c2 h z−z ′ 1 (2.2.15) r (h) where the matrix norm in the left side (recall that ∂ g (x, y) is a 2 × 2 matrix) is the max norm, i.e., the maximum over the matrix elements, 2 2 r ∂ r := i,j=1 ∂i,ji,j , r! = i,j=1 ri,j !, and z1 = |perL (z1 )| + |z2 |, see (2.2.12), Moreover, if z, z ′ ∈ Λ are such that |perL (z1 − z1′ )| < L/2 − |r1,1 | − |r2,1 |, 2. (h) ∂ r gE (z, z ′ ) ≤ C 1+|r |1 r!2(1+|r |1 )h e−c2 h dE (z,z ′ ) (2.2.16) where dE (z, z ′ ) := min{|perL (z1 − z1′ )| + min{z2 + z2′ , 2(M + 1) − z2 − z2′ }, L − |perL (z1 − z1′ )| + |z2 − z2′ |}. Finally, there exists a Hilbert space HLM with inner product (·, ·) including (h) (h) (≤h) (≤h) elements γω,s,z , γ̃ω,s,z , γω,s,z , γ̃ω,s,z (for s = (s1 , s2 ) ∈ Z2+ , x ∈ Λ) such that whenever h∗ ≤ h ≤ 0,   ′ (h) (h) (h) 3. ∂ (s,s ) gωω′ (z, z ′ ) ≡ γ̃ω,s,z , γω′ ,s ′ ,z′ and   ′ (≤h) (≤h) (≤h) ∂ (s,s ) gωω′ (z, z ′ ) ≡ γ̃ω,s,z , γω′ ,s ′ ,z′ , and          (h) 2  (h) 2  (≤h) 2  (≤h) 2 4. γω,s,z  , γ̃ω,s,z  , γω,s,z  , γ̃ω,s,z  ≤ C 1+|s|1 s!2h(1+2|s|1 ) where | · | is the norm generated by the inner product (·, ·). Combining points 3 and 4, we see that Corollary 2.4. For all z, z ′ ∈ Λ, r ∈ Z4+ , and h∗ ≤ h ≤ 0, ∂ r g(≤h) (z, z ′ ) ≤ C 1+|r |1 r!2(1+|r |1 )h . (h) (h) (2.2.17) Remark 2.5. Since gB = g(h) − gE , from items 1 and 2 it follows that the (h) bulk, single-scale, propagator gB satisfies the same estimate as (2.2.15) for all x, y ∈ Λ allowed in Item 2, i.e., whenever |perL (z1 − z1′ )| < L/2 − |r1,1 | − |r2,1 |. The latter restriction on z, z ′ just comes from the requirement that the discrete derivatives do not act on the discontinuous functions sL and perL entering the (h) (h) definition of gB (and, therefore, of gE ); in fact, one can easily check from the proof that, if r = 0, then (2.2.16) is valid for all z, z ′ ∈ Λ, without further restrictions. 1078 G. Antinucci et al. Ann. Henri Poincaré Remark 2.6. All the estimates stated in the proposition are uniform in L, M , therefore, they remain valid for the L, M → ∞ limit of the propagators. In (h) (≤h) particular, g∞ (z − z ′ ) and g∞ (z − z ′ ) satisfy the same estimates as (2.2.15) and (2.2.17), respectively. Similarly, the Gram representation stated in items (h) 3 and 4 is also valid for g∞ . Therefore, if |z1 − z1′ | ≤ L/2 − |s1 | − |s′1 |, also ′ ′ ′ (h) (h) ∂ (s,s ) gE (x, y) = ∂ (s,s ) g(h) (z, z ′ ) − ∂ (s,s ) g∞ (z − z ′ ) admits a Gram representation, with qualitatively the same Gram bounds (and, of course, such a representation can be extended by anti-periodicity to all z, z ′ such that |perL (z1 − z1′ )| ≤ L/2 − |s1 | − |s′1 |). Remark 2.7. If we rename the massive propagator in (2.1.16) as gm (z, z ′ ) =: (1) (1) (1) g(1) (z, z ′ ) and then use it to define g∞ , gB , and gE in the same way that we did above with g[η] , it is straightforward to check that the estimates in items 1 and 2 of the proposition remain valid for h = 1. Similarly, the reader can check that the proof of items 3 and 4 given below can be straightforwardly applied to the case h = 1, as well. Remark 2.8. Again since D(k1 , k2 ) is exactly the denominator in the definition (2.1.20), it is easily seen that fη (k1 , k2 )ĝ(k1 , k2 ) is an entire function of t1 . All of the bounds in Proposition 2.3 are obtained by writing the relevant quantities as absolutely convergent integrals or sums in k1 , k2 , and η; since these bounds are locally uniform in t1 as long as it is bounded away from 0 and 1, we also see that all of the propagators are analytic functions of t1 with all other arguments held fixed. The proof of Proposition 2.3 is in Appendix B. 2.3. Asymptotic Behavior of the Critical Propagator Although our main result, Theorem 1.1, involves correlation functions for a finite lattice, since the continuum limit of the energy correlation functions of the non-interacting model is well understood [11,24], as a result we also obtain a characterization of the scaling limit [4, Corollary 1.2]. For completeness, we give here a description of the scaling limit of the critical propagator, from which the non-interacting energy correlation functions are easily calculated. We rescale the lattice as follows: fix two positive constants ℓ1 , ℓ2 (no condition on the ratio ℓ1 /ℓ2 ), and let L = 2⌊a−1 ℓ1 /2⌋, M = ⌊a−1 ℓ2 ⌋ for a > 0 the lattice mesh. Let Λa := aΛ and Λℓ1 ,ℓ2 the continuum cylinder. We also let  ·  indicate the Euclidean distance on the cylinder Λℓ1 ,ℓ2 . Given z, z ′ ∈ Λℓ1 ,ℓ2 , we let gc,a (z, z ′ ) = a−1 gc (⌊a−1 z⌋, ⌊a−1 z ′ ⌋). (2.3.1) The main result of this section concerns the limiting behavior of gc,a as a → 0. Proposition 2.9. Given ℓ1 , ℓ2 > 0, there exist C, c > 0 such that for all x, y ∈ Λℓ1 ,ℓ2 such that z = z ′ and a > 0 for which a(min{ℓ1 , ℓ2 , z − z ′ })−1 ≤ c, gc,a (z, z ′ ) = gscal (z, z ′ ) + Ra (z, z ′ ), (2.3.2) Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry  and Ra (z, z ′ ) ≤ Ca min{ℓ1 , ℓ2 , z − z ′ } ′ gscal (z, z ) =  (−1) n scal g∞ (z1 − ′ z1 −2 + n1 ℓ1 , z2 − 1079 , where ′ z2 + 2n2 ℓ2 ) n ∈Z2 +  g2scal (z1 − z1′ + n1 ℓ1 , z2 + z2′ + 2n2 ℓ2 ) −g1scal (z1 − z1′ + n1 ℓ1 , z2 + z2′ + 2n2 ℓ2 ) −g2scal (z1 − z1′ + n1 ℓ1 , z2 + z2′ + 2n2 ℓ2 ) g1scal (z1 − z1′ + n1 ℓ1 , z2 + z2′ + 2(n2 − 1)ℓ2 ) ! , (2.3.3) and where, letting g scal  dk1 dk2 −ik1 z1 −ik2 z2 −ik1 1 e (z1 , z2 ) := t2 (1 − t2 ) (2π)2 k12 + k22 2 R z1 1 = − , 2πt2 (1 − t2 ) z12 + z22 (2.3.4) z2 z1 we denoted g1scal (z1 , z2 ) := g scal ( 1−t , z2 ), g2scal (z1 , z2 ) := g scal ( 1−t , z1 ), 2 1−t1 1 1−t2 and  scal  g1 (z1 , z2 ) g2scal (z1 , z2 ) scal g∞ (z1 , z2 ) := scal . (2.3.5) g2 (z1 , z2 ) −g1scal (z1 , z2 ) The proof of Proposition 2.9 is given in Appendix C. It is easy to see from the definition of gscal that its entries vanish for z2 = 0 and/or z2 = ℓ2 c and/or z2′ = 0 and/or z2′ = ℓ2 in a fashion analogous to the one discussed in Remark 2.2. 2.4. Symmetries of the Propagator Note that the action St1 ,t2 (Φ) of Eq. (2.1.2) is unchanged by the substitutions H z → iHθ1 z , Hz → iH θ1 z , with θ1 (z1 , z2 ) := (L + 1 − z1 , z2 ), or H z → −iH θ2 z , Hz → iHθ2 z , V z → iV θ1 z , V z → iVθ2 z , Vz → −iVθ1 z (2.4.1) Vz → iV θ2 z (2.4.2) where θ2 (z1 , z2 ) := (z1 , M + 1 − z2 ). These transformations, of course, correspond to the reflection symmetries of the Ising model on a cylinder. In terms of the φ, ξ variables, it is easy to see from Eq. (2.1.10) that these substitutions are equivalent to φ±,z → Θ1 φ±,z := ±iφ±,θ1 z , ξ±,z → Θ1 ξ±,z := iξ∓,θ1 z (2.4.3) ξ±,z → Θ2 ξ±,z := ∓iξ±,θ2 z . (2.4.4) and φ±,z → Θ2 φ±,z := iφ∓,θ2 z , With a little more notation we can write this more compactly: letting φω,z denote φ+,x , φ−,x , ξ+,x , ξ−,x for ω = 1, −1, +i, −i, respectively, and letting θj ω := (−1)j+1 ω for ω ∈ C, Eqs. (2.4.3) and (2.4.4) can be combined into Θj φω,z := iαj,ω φθj ω,θj x (2.4.5) where αj,ω is −1 if (j = 1 and ω = −1) or (j = 2 and ω = i), and 1 otherwise. Since these transformations act on the vector Ψ as orthogonal matrices, this is equivalent to the symmetry of the coefficient matrix A (and therefore 1080 G. Antinucci et al. Ann. Henri Poincaré its inverse) under the associated similarity transform, and since gc is just a diagonal block of A−1 we have   −gc;++ gc;+− gc (z, z ′ ) = (2.4.6) (θ1 z, θ1 z ′ ) gc;−+ −gc;−− and   gc;−− gc;−+ gc (z, z ) = − (θ2 z, θ2 z ′ ) . gc;+− gc;++ ′ (2.4.7) These relationships can also be recovered from Eq. (2.1.19), using the observations on ĝ∞ in Remark 2.1. This latter point is helpful because, since fη is even [η] in both k1 and k2 , it also applies to gc . Taking the appropriate L, M → ∞ limit we also obtain # # " [η] " [η] [η] [η] g∞;−− g∞;−+ −g∞;++ g∞;+− [η] (z1 , −z2 ) , (−z1 , z2 ) = − [η] g∞ (z1 , z2 ) = [η] [η] [η] g∞;+− g∞;++ g∞;−+ −g∞;−− (2.4.8) [η] gB has the symmetries (2.4.6) and (2.4.7). Applying which also implies that the differences and integrals in the relevant definitions we see that (h) (≤h) (h) (h) Lemma 2.10. gc , gc , gB , and gE all have the symmetries (2.4.6) (h) and (2.4.7) for any h∗ ≤ h ≤ 0, and g∞ has the symmetries (2.4.8) for any h ≤ 0. (1) For gc ≡ gm we have     gm;−− gm;−+ −gm;++ gm;+− ′ ′ (θ1 z, θ1 z ) = (θ2 z, θ2 z ′ ) gm (z, z ) = − gm;+− gm;++ gm;−+ −gm;−− (2.4.9) (1) (1) (1) which similarly extends to g∞ , gB , and gE . 3. Grassmann Representation of the Generating Function In this section we rewrite the generating function of the energy correlations for the Ising model (1.1) with finite range interactions as an interacting Grassmann integral, and we set the stage for the multiscale integration thereof, to be discussed in the following sections. The estimates in this and in the following section are uniform for J1 /J2 , L/M ∈ K and t1 , t2 ∈ K ′ , but may depend upon the choice of K, K ′ , with K, K ′ the compact sets introduced before the statement of Theorem 1.1. As anticipated there, we will think of K, K ′ as being fixed once and for all and, for simplicity, we will not track the dependence upon these sets in the constants C, C ′ , . . . , c, c′ , . . . , κ, κ′ , . . ., appearing below. Unless otherwise stated, the values of these constants may change from line to line. Our goal is to show that the generating function (1.6) of the energy correlations can be replaced, for the purpose of computing multipoint energy Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1081 correlations at distinct edges, by the Grassmann generating function (1.7). Our main representation result for the Taylor coefficients at A = 0 of the logarithm of ZΛ (A), analogous to [18, Proposition 1], is the following. Proposition 3.1. For any translation invariant interaction V of finite range, there exists λ0 = λ0 (V ) such that, for any |λ| ≤ λ0 (V ),     ∂ ∂ ∂ ∂ $  ... log ZΛ (A) ... log ΞΛ (A) = ∂Ax ∂Ax ∂Ax ∂Ax 1 A =0 n 1 A =0 n as long as n ≥ 2 and the xj are distinct, where  $ Λ (A) := eW(A ) DΦ eSt1 ,t2 (Φ)+V(Φ,A ) Ξ (3.1) where St1 ,t2 was defined in (2.1.2) and, recalling that Ex is the Grassmann binomial defined after (2.1.2): 1.     V(Φ, A) = (1 − t2j(x) )Ex Ax + WΛint (X, Y ) Ex Ax X,Y ⊂BΛ X =∅ x∈BΛ x∈X x∈Y (3.2) ≡ Bfree (Φ, A) + V int (Φ, A) where, for any n ∈ N, m ∈ N0 , and suitable positive constants C, c, κ,  |WΛint (X, Y )|ecδ(X∪Y ) ≤ C m+n |λ|max(1,κ(m+n)) sup x0 ∈BΛ X,Y ⊂BΛ : x0 ∋X |X|=n, |Y |=m (3.3) and δ(X), for X ⊂ BΛ , denotes the size of the smallest Z ⊃ X which is the edge set of a connected subgraph of GΛ . 2. W(A) =  Y ⊂BΛ |Y |≥2 wΛ (Y )  Ax where, for any m ∈ N, and the same C, c, κ as above,  |wΛ (Y )|ecδ(Y ) ≤ C m |λ|max(1,κm) . sup x0 ∈BΛ (3.4) x∈Y (3.5) Y ⊂BΛ : x0 ∋Y, |Y |=m 3. WΛint , wΛ , considered as functions of λ, t1 , and t2 , can be analytically continued to any complex λ, t1 , t2 such that |λ| ≤ λ0 and |t1 |, |t2 | ∈ K ′ , with K ′ the same compact set introduced before the statement of Theorem 1.1, and the analytic continuations satisfies the same bounds above. Proof. The proof is basically the same as [18, Proposition 1], so we refer to that for the details. Note that the restriction in [18] to a pair interaction is unimportant, since any even interaction of the form V (X)σX with X ⊂ Λ can always be written as a product of factors of ǫx , in analogy with the rewriting σz σz′ = 21 (Uz,z′ + Dz,z′ ) discussed after [18, Equation (2.8)]; the only difference 1082 G. Antinucci et al. Ann. Henri Poincaré in the current setting is that the ‘strings’ graphically associated with Uz,z′ and Dz,z′ , see [18, Figure 3], are replaced by other figures, whose specific shape depends on V (X) and that one should use t1 or t2 in place of t as appropriate5 . Note that the set of strings associated with a pair interaction, or the set of more general figures associated with a generic even interaction, is, or can be chosen to be, invariant under the basic symmetries of the model, namely horizontal translations, and horizontal and vertical reflections; therefore, in the following, we shall assume that such a graphical representation is invariant under these symmetries. By proceeding as in [18] we get the analogue of [18, Eq.(2.20)], namely      WΛ (X, Y ) Ex ϕT (Γ) Ax ζG (γ) = W(A) + V(Φ, A) = Γ∈CΛ γ∈Γ X,Y ⊂BΛ x∈X x∈Y (3.6) where CΛ is the set of multipolygons in Λ, ϕT is the Mayer’s coefficient, and ζG is the activity of the polygon γ, which is a polynomial in the Ex , Ax for edges x in γ (for more details about the notation and more precise definitions, we refer to [18]). The terms with X = ∅ contribute to W(A) (that is, we let wΛ (Y ) := WΛ (∅, Y )), while those with X = ∅ contribute to V(Φ, A) (note $ Λ (A) of order 2 or that, for the purpose of computing the derivatives of log Ξ more, the terms with |X| = 0 and |Y | = 0, 1 can be dropped from the definition of W(A), and we do so). The explicit computation of the term independent of λ, which has X = Y ∈ B, leads to the decomposition in Eq. (3.2). The bounds Eqs. (3.3) and (3.5) follow directly from the bounds in [18], see, e.g., [18, Eq.(2.25)] and following discussion. Finally, the analyticity property (which was used implicitly in [18]) follows by noting that we have defined all of the quantities of interest as uniformly absolutely convergent sums of terms which are themselves analytic functions of λ, t1 , t2 as long as the absolute values of these parameters belong to the appropriate intervals.  With a view toward the analysis of finite size effects in [4] (and, in particular, toward the claims done in [4, Section 2.2] after the statement of [4, Proposition 2.5]), it is convenient to decompose the kernel WΛint of V int (Φ, A) into a ‘bulk’ plus an ‘edge’ part. This requires a bit of notation. Note that any subset X of Λ with horizontal diameter smaller than L/2 can be identified (non-uniquely, of course) with a subset of Z2 with the same diameter and ‘shape’ as X; we call X∞ ⊂ Z2 one of these arbitrarily chosen representatives6 5 We take the occasion to point out that [18, Figure 4] contains a mistake: the string S2 depicted there is not allowed by the conventions of [18]: the shape S2 can only be obtained as the union of two appropriate strings. 6 For instance, given X = {z , . . . , z }, recalling that (z ) ∈ {1, 2, . . . , L} and (z ) ∈ n 1 i 1 i 2 {1, 2, . . . , M }, we can let X∞ = {y1 , . . . , yn } be the set of points such that: (1) the vertical coordinates are the same as those of z , i.e., (yi )2 = (zi )2 , ∀i = 1, . . . , n; (2) the horizontal coordinate of y1 is the same as z1 , i.e., (y1 )1 = (z1 )1 ; (3) all the other horizontal coordinates are the same modulo L, i.e., (yi )1 = (zi )1 mod L, ∀i = 2, . . . , n; (4) the specific values of (yi )2 Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1083 of X, and we shall use an analogous convention for the subsets of BΛ with horizontal diameter smaller than L/2. Lemma 3.2. Under the same assumptions of Proposition 3.1, the kernel WΛint of V int (Φ, A) can be decomposed as int WΛint (X, Y ) = W∞ (X∞ , Y∞ ) 1(diam1 (X ∪ Y ) ≤ L/3) + WEint (X, Y ) =: WBint (X, Y ) + WEint (X, Y ), (3.7) where diam1 is the horizontal diameter on the cylinder Λ; X∞ , Y∞ ⊂ B := BZ2 are two representatives of X, Y , respectively, such that X∞ ∪ Y∞ is a representative of X ∪Y , in the sense defined before the statement of the lemma; int is a function, independent of L, M , invariant under translations and W∞ under reflections about either coordinate axis, which satisfies the same weighted L1 bound (3.3) as WΛint . Moreover, for any n ∈ N and m ∈ N0 , WEint satisfies  1 |WEint (X, Y )|ecδE (X∪Y ) ≤ C m+n |λ|max(1,κ(m+n)) , (3.8) L X,Y ⊂BΛ |X|=n, |Y |=m with the same C, c, κ as in Proposition 3.1, where δE (X) is the cardinality of the smallest connected subset of BΛ which includes X and either touches the boundary of the cylinder7 , or its horizontal diameter is larger than L/3. Proof. In order to obtain the decomposition (3.7), let int W∞ (X∞ , Y∞ ) := lim L,M →∞ WΛint (X∞ + zL,M , Y∞ + zL,M ), (3.9) where zL,M = (L/2, ⌊M/2⌋) and X∞ + zL,M is the translate of X∞ by zL,M ; note that this limit is well defined thanks to the fact that WΛint can be expressed in terms of a sum like [18, Eq.(2.21)], which is exponentially convergent, see int satisfies the analogue of (3.6), that is [18, Eq.(2.25)]. The kernel W∞  Γ∈C∞ T ⎛ ϕ (Γ) ⎝  γ∈Γ ζG (γ) −  γ∈Γ ⎞  ζG (γ)λ=0 ⎠ =  X,Y ⊂B int W∞ (X, Y )  x∈X Ex  Ax , x∈Y (3.10) 2 with C∞ the set of multipolygons on Z , and the activity ζG (γ) the same as the one in (3.6), provided that γ is considered now as a polygon in Z2 , rather than in Λ (note that such identification is possible as long as γ does not wrap int around the cylinder). Moreover, W∞ is translation invariant, and, letting int WEint (X, Y ) := WΛint (X, Y ) − 1(diam1 (X ∪ Y ) ≤ L/3)W∞ (X∞ , Y∞ ), (3.11) the contribution to the first term in the right side of all multipolygons in CΛ with horizontal diameter ≤ L/3 cancels with their counterparts in C∞ from for i ≥ 2 are chosen in such a way that the horizontal distances between the corresponding pairs in X and X∞ are the same, if measured on the cylinder Λ or on Z2 , respectively. 7 We say that X ⊂ B ‘touches the boundary of the cylinder Λ’, if at least one of the edges Λ in X has an endpoint whose vertical coordinate is either equal to 1 or to M . 1084 G. Antinucci et al. Ann. Henri Poincaré the second term in the right side. Each of the remaining multipolygons either comes from the first term in the right side and involves a multipolygon in CΛ whose support has horizontal diameter larger than L/3, or comes from the second term in the right side and involves a multipolygon in C∞ whose support contains a set Z∞ which is the representative (in the sense explained before the statement of the lemma) of a connected subset ZΛ of BΛ that contains X ∪Y and touches the boundary of Λ; in either case the number of edges in the support of such a multipolygon is at least δE (X ∪ Y ), from which the bound (3.8) follows.  In the following, we will wish to work in the φ, ξ variables introduces in Sect. 2.1.2; applying the change of variables (2.1.10), with some abuse of notation we rewrite V(Φ, A) in (3.2) as:   V(φ, ξ, A) = BΛfree ((ω, z), x)φ(ω, z)Ax x∈BΛ (ω ,z )∈O 2 ×Λ2 +    WΛint ((ω, z), x)φ(ω, z)A(x) n∈2N (ω ,z )∈O n ×Λn x∈Bm Λ m∈N0 =: B free (φ, ξ, A) + V int (φ, ξ, A), (3.12) where N and N0 are the sets of positive ad nonnegative integers, respectively, O := {1, −1, i, −i}, and, for ω ∈ On , z ∈ Λ, we denote φ(ω, z) = n  φωj ,zj (3.13) j=1 m with φ±i,z = ξ±,z , and similarly A(x) := j=1 Axj (for x = ∅, we interpret A(∅) = 1). The decay properties of s± noted after Eq. (2.1.11) together with Eq. (3.3) imply a similar decay property for the new coefficients:    (3.14) sup sup ecδ(z ,x) BΛfree ((ω, z), x) ≤ C, ω ∈O 2 x∈BΛ z ∈Λ2 and sup sup ω ∈O n z1 ∈Λ sup  z2 ,...,zn ∈Λ sup ω ∈O n x1 ∈BΛ    ecδ(z ) WΛint ((ω , z ), ∅) ≤ C n |λ|max(1,κn)  x2 ,...,xm ∈BΛ z ∈Λn   ecδ(z ,x ) WΛint ((ω , z ), x) ≤ C n+m |λ|max(1,κ(n+m)) . (3.15) Note that, with this rewriting in terms of the φ, ξ variables, recalling that St1 ,t2 (Φ) = Sm (ξ) + Sc (φ), see (2.1.12)–(2.1.13), and denoting Pc (Dφ) := Dφ eSc (φ) / Pf(Ac ), Pm (Dξ) := Dξ eSm (ξ) / Pf(Am ) (here Ac and Am are the Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1085 two 2|Λ| × 2|Λ| anti-symmetric matrices associated with the Grassmann quadratic forms Sc (φ) and Sm (ξ), respectively), the Grassmann generating func$ Λ (A) in (3.1) can be rewritten as tion Ξ   $ Λ (A) ∝ eW(A ) Pc (Dφ) Pm (Dξ)eV(φ,ξ,A ) , Ξ (3.16) where ∝ means ‘up to a multiplicative constant independent of A’. In view of these rewritings, Proposition 3.1 implies [4, Proposition 2.5] as an immediate corollary. Of course, the bulk–edge decomposition of Lemma 3.2 implies an analogous decomposition for the kernel of V int (φ, ξ, A), which reads as follows: int ((ω, z ∞ ), x∞ ) WΛint ((ω, z), x) = (−1)α(z ) 1(diam1 (z, x) ≤ L/3) W∞ + WEint ((ω, z), x) =: WBint ((ω, z), x) + (3.17) WEint ((ω, z), x), where, for any z with diam1 (z) ≤ L/3, α(z) = #{zi ∈ z : (zi )1 ≤ L/3}, 0, if maxzi ,zj ∈z {(zi )1 − (zj )1 } ≥ 2L/3, otherwise. (3.18) and int W∞ ((ω, z), x) := lim L,M →∞ WΛint ((ω, z + zL,M ), x + zL,M ). (3.19) The factor (−1)α(z ) in front of the first term in the right side of (3.17), in light int of the antiperiodicity of the φ, ξ fields, guarantees that W∞ is translation invariant (in both coordinate directions), and that both WBint and WEint are invariant under simultaneous translations of z and x in the horizontal direction, with anti-periodic and periodic boundary conditions in z and x, respectively. In terms of this new notation, Eq.(3.8) implies that, for any n ∈ N and m ∈ N0 ,  1 sup |WEint ((ω, z), x)|ecδE (z ,x) ≤ C m+n |λ|max(1,κ(m+n)) , (3.20) L ω ∈On n z ∈Λ x∈Bm Λ with δE (z, x) is the ‘edge’ tree distance of (z, x), i.e., the cardinality of the smallest connected subset of BΛ that includes x, touches the points of z and either touches the boundary of the cylinder or it has horizontal diameter larger than L/3. Of course, BΛfree admits a similar bulk–edge decomposition: free + BEfree with in analogy with (3.7), BΛfree = BB  1 sup |BEfree (ω, z, x)|ecδE (z ,x) ≤ C. (3.21) L ω ∈O2 2 z ∈Λ x∈BΛ Before concluding this section, let us comment on the connection between (3.16) and (1.7). Fix once and for all a neighborhood U ⊂ R of 1 not containing 0; say, for definiteness, U := {z ∈ R : |z − 1| ≤ 1/2}. For any Z ∈ U and 1086 G. Antinucci et al. Ann. Henri Poincaré ∗ t∗1 ∈ K ′ , we let t∗2 := (1−t∗1 )/(1+t∗1 ) and let Sc∗ (φ) = 12 (φ, A∗c φ) (resp. Sm (ξ) = 1 ∗ ∗ ∗ ξ)) be obtained from S (resp. S ) by replacing t , t with t , t2 in (ξ, A c m 1 2 m 1 2 ∗ Eq. (2.1.13) (resp. Eq. (2.1.12)). We also let Pc∗ (Dφ) := DφeSc (φ) / Pf(A∗c ), ∗ ∗ (Dξ) := DξeSm (ξ) / Pf(A∗m ). Given these definitions, in (3.16) we first rescale Pm the φ and ξ variables by Z −1/2 , then multiply and divide the Grassmann ∗ ∗ integrand by eSc (φ)+Sm (ξ) , thus getting  (1) W(A ) ∗ $ Ξ(A) ∝ e Pc∗ (Dφ)Pm (Dξ)eV (φ,ξ,A ) (3.22) with ∗ V (1) (φ, ξ, A ) := Z −1 Sc (φ) − Sc∗ (φ) + Z −1 Sm (ξ) − Sm (ξ) + V(Z −1/2 φ, Z −1/2 ξ, A ) =: Nc (φ) + Nm (ξ) + V(Z −1/2 φ, Z −1/2 ξ, A ). (3.23) This proves (1.7) and puts us in the position of setting the multiscale computation of the sequence of effective potentials, whose infinite plane counterparts are constructed and bounded in the next section. For later reference, we note that, in light of (2.1.12), (2.1.13), (3.12), V (1) (φ, ξ, A) can be written as:    (1) V (1) (φ, ξ, A) = WΛ ((ω, z), x)φ(ω, z)A(x), n∈2N, (ω ,z )∈O n ×Λn x∈Bm Λ m∈N0 (3.24) for an appropriate kernel, which inherits its properties from those of Sc , (1) Sm and V. With no loss of generality, we can assume that WΛ is antisymmetric under simultaneous permutations of ω and z, symmetric under permutations of x, invariant under simultaneous translations of z and x in the horizontal direction (with anti-periodic and periodic boundary conditions in z and x, respectively), invariant under the reflection symmetries induced by the transformations Ax → Aθl x and φω,z → Θl φω,z , see (2.4.3)–(2.4.4). From now on, with some abuse of notation, given ω = (ω1 , . . . , ωn ) ∈ On and z = (z1 , . . . , zn ) ∈ Λn , we shall identify the pair (ω, z) with the n-ple ((ω1 , z1 ), . . . , (ωn , zn )) ∈ (O × Λ)n . 4. The Renormalized Expansion in the Full-Plane Limit In this section we construct the sequence of effective potentials (see the last part of Sect. 1) in the infinite volume limit and derive weighted L1 bounds for their kernels, in a form appropriate for the subsequent generalization to the finite cylinder, discussed in [4, Section 3]. The construction of this section will allow us, in particular, to fix the free parameters β, Z, t∗1 , which the Grassmann integral in the right side of (3.22) depends on, in such a way that the sequence of running coupling constants goes to zero exponentially fast in the infrared limit; see Sect. 4.5 below. As anticipated at the end of Sect. 1, here we limit ourselves to construct the sequence of effective potentials at A = 0, so, for lightness of notation, we denote by V (h) (φ) := V (h) (φ, 0) the effective potentials with h ≤ 0 at zero Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1087 external fields (similarly, we let V (1) (φ, ξ) = V (1) (φ, ξ, A)). In light of (1.8) and (1.10), these effective potentials are iteratively defined via ) ∗ (1) log Pm (Dξ) eV (φ,ξ) if h = 1, (h−1) ) (h) V (φ) = const. + LV (h) (φ+ϕ)+RV (h) (φ+ϕ) log P (Dϕ) e if h ≤ 0, (4.1) where the const. is fixed so that V (h) (0) = 0, for all h ≤ 0, and LV (h) + RV (h) is an equivalent rewriting of V (h) , to be defined (in the full plane limit) below. Expanding the exponential and the logarithm in the right side of (4.1) allows us to rewrite ⎞ ⎛  1 (4.2) E∗ ⎝V (1) (φ, ·); · · · ; V (1) (φ, ·)⎠ V (0) (φ) = const. + +, s! m * s≥1 and, for h ≤ 0, V (h−1) (φ) = const. + ⎛ s times  1 s! s≥1 ⎞ E(h) ⎝LV (h) (φ + ·) + RV (h) (φ + ·); · · · ; LV (h) (φ + ·) + RV (h) (φ + ·)⎠ , +, * s times (4.3) E∗m (h) where (resp. E ) denotes the truncated expectation [15, Eq.(4.13)] with ∗ respect to the Grassmann Gaussian integration Pm (resp. P (h) ). Expanding the effective potentials in terms of their kernels, in analogy with (3.24), Eqs. (4.2)–(4.3) allow us to iteratively compute the kernels of V (h) , for all h ≤ 0. For instance, at the first step, using (3.24), (4.2) and the BBFK formula (for Battle, Brydges, Federbush, Kennedy) for the Grassmann truncated (1) (1) expectations [1,7–9], we find that, denoting by VΛ (Ψ) = WΛ (Ψ, ∅) with n Ψ = ((ω1 , z1 ), . . . , (ωn , zn )) ∈ ∪n∈2N (O × Λ) =: M1,Λ the kernel of V (1) (φ, ξ), (0) the kernel VΛ of V (0) (φ) satisfies, for any Ψ = ((ω1 , z1 ), . . . , (ωn , zn )) ∈ ∪n∈2N ({+, −} × Λ)n , (0) VΛ (Ψ) ∞  1 = s! s=1 (Ψ)   (1) GT (Ψ̄1 , . . . , Ψ̄s ) Ψ1 ,...,Ψs ∈M1,Λ T ∈S(Ψ̄1 ,...,Ψ̄s ) ⎛ α(Ψ; Ψ1 , . . . , Ψs ) ⎝ s  j=1 ⎞ (1) VΛ (Ψj )⎠ , (4.4) where • the symbol (Ψ) on the second sum means that the sum runs over all ways of representing Ψ as an ordered sum of s (possibly empty) tuples, Ψ′1 ⊕ · · · ⊕ Ψ′s = Ψ, and over all tuples M1,Λ ∋ Ψj ⊇ Ψ′j (here ⊕ indicates concatenation of ordered tuples); for each such term in the second sum, 1088 G. Antinucci et al. Ann. Henri Poincaré we denote by Ψ̄j := Ψj \ Ψ′j and by α(Ψ; Ψ1 , . . . , Ψs ) the sign of the permutation from Ψ1 ⊕ · · · ⊕ Ψs to Ψ ⊕ Ψ̄1 ⊕ · · · ⊕ Ψ̄s (here ⊕ indicates concatenation of ordered tuples); • S(Ψ̄1 , . . . , Ψ̄s ) denotes the set of all the ‘spanning trees’ on Ψ̄1 , . . . , Ψ̄s , that is, of all the sets T of ordered pairs (f, f ′ ), with f ∈ Ψ̄i , f ′ ∈ Ψ̄j and i < j, whose corresponding graph GT = (V, ET ), with vertex set V = {1, . . . , s} and edge set ET = {(i, j) ∈ V 2 : ∃(f, f ′ ) ∈ T with f ∈ Qi , f ′ ∈ Qj }, is a tree graph (for s = 1, we let S(Ψ̄1 ) ≡ {∅}); (1) • GT (Ψ̄1 , . . . , Ψ̄s ) is different from zero only if Ψ̄j ∈ ∪2N0 ({+i, −i} × Λ)n for all j = 1, . . . , s, and, if s > 1, only if Ψ̄j = ∅ for all j = 1, . . . , s; more (1) precisely: if s = 1 and Ψ̄1 = ∅, then G∅ (∅) = 1; if s = 1 and Ψ̄1 = ∅,  (1) (1) then G∅ (Ψ̄1 ) = Pf GΨ̄1 , where, given a pair ℓ = ((ω, z), (ω ′ , z ′ )) of  (1) (1) (1) distinct elements of Ψ̄1 , GΨ̄1 ℓ = gℓ := g−iω,−iω′ (z, z ′ )8 ; if s > 1 and Ψ̄j = ∅ for all j = 1, . . . , s, then (1) GT (Ψ̄1 , . . . , Ψ̄s ) := αT (Ψ̄1 , . . . , Ψ̄s ) "  (1) gℓ ℓ∈T #  (1) PΨ̄1 ,...,Ψ̄s ,T ( dt) Pf GΨ̄1 ,...,Ψ̄s ,T (t) , (4.5) where – αT (Ψ̄1 , . . . , Ψ̄n ) is the sign of the permutation from Ψ̄1 ⊕ · · · ⊕ Ψ̄s to T ⊕ (Ψ̄1 \ T ) ⊕ · · · ⊕ (Ψ̄s \ T ); – t = {ti,j }1≤i,j≤s , and PΨ̄1 ,...,Ψ̄s ,T ( dt) is a probability measure with support on a set of t such that ti,j = ui · uj for some family of vectors ui = ui (t) ∈ Rs of unit norm; s (1) – letting 2q = i=1 |Ψ̄i |, GΨ̄1 ,...,Ψ̄s ,T (t) is an antisymmetric (2q − 2s + 2) × (2q − 2s + 2) matrix, whose off-diagonal elements are given  (1) (1) by GΨ̄1 ,...,Ψ̄s ,T (t) f,f ′ = ti(f ),i(f ′ ) gℓ(f,f ′ ) , where f, f ′ are elements of the tuple (Ψ̄1 \ T ) ⊕ · · · ⊕ (Ψ̄s \ T ), and i(f ) is the integer in {1, . . . , s} such that f is an element of Ψ̄i \ T . (1) Recalling that gℓ (1) and VΛ (1) (1) admit infinite volume limits gℓ,∞ and V∞ , respec(0) tively, in the sense of (2.2.9) and (3.19), from (4.4) it follows that VΛ admits an infinite volume limit as well, equal to the ‘obvious’ analogue of the right side of (4.4), namely, the expression obtained from that one by replacing: M1,Λ by (1) (1) M1,∞ := ∪n∈2N (O × Z2 )n ; GT by GT,∞ (the latter being defined analogously (1) to the former, with gℓ (1) VΛ (1) by V∞ . (1) gσ,σ′ (z, z ′ ) with σ, σ ′ ∈ {+, −} ∗ ′ gm (z, z ), where g∗m (z, z ′ ) the same as 8 Here (1) replaced by gℓ,∞ in all the involved expressions); and are the elements of the 2 × 2 matrix g(1) (z, z ′ ) ≡ in (2.1.16) with t1 replaced by t∗1 . Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1089 Proceeding inductively in h ≤ 0, one finds that (4.3) implies a repre(h) sentation of the kernel VΛ of V (h) analogous to (4.4). Also in that case, (h) the resulting formula for VΛ admits a natural infinite volume limit. In this (h) way, we obtain a recursive equation for the infinite plane kernels, denoted V∞ , whose definition and solution is described below. Convergence of the finite vol(h) ume kernels VΛ to their infinite volume counterparts, with optimal bounds on the norm of the finite size corrections, is deferred to [4, Section 3]. A key point in the derivation of bounds on the kernels that are uniform in the scale label h is the definition of an appropriate action of the L and R operators, as well as of their infinite volume counterparts, L∞ and R∞ . As anticipated above, these operators allow us to isolate the potentially divergent part of the kernels, LV (h) (the ‘local’ contributions, parametrized at any given scale, by a finite number of ‘running coupling constants’) from a remainder RV (h) , which is ‘dimensionally better behaved’ than LV (h) ; in order for the remainder to be shown to satisfy ‘improved dimensional bounds’, it is necessary to rewrite it in an appropriate, interpolated, form, involving the action of discrete derivatives on the Grassmann fields. The plan of the incoming subsections is the following: in Sect. 4.1 we describe the representation of the effective potentials in the infinite volume limit and introduce the notion of equivalent kernels; in Sect. 4.2 we define the operators L∞ and R∞ ; in Sect. 4.3 we derive the solution to the recursive equations for the infinite volume kernels in terms of a tree expansion; in Sect. 4.4, we use such a tree expansion to derive weighted L1 bounds on the kernels; importantly, these bounds depend upon the sequence of running coupling constants, and they imply analyticity of the kernels provided such a sequence is uniformly bounded in the scale label; in Sect. 4.5, as a corollary of the weighted L1 bounds of the previous subsection, we prove a fixed point theorem, which allows us to fix the free parameters Z, β, t∗1 in such a way that the flow of the running coupling constants is, in fact, uniformly bounded in h, as desired: even more, the running coupling constants go to zero exponentially fast as h → −∞, a consequence of the irrelevance of the quartic effective interaction in the theory at hand. 4.1. Effective Potentials and Kernels: Representation and Equivalence In this subsection we define the effective potential in the full plane in terms of equivalence classes of kernels V (Ψ), namely, of real-valued functions play(0) (1) ing the same role as the coefficient functions V∞ (Ψ) and V∞ (Ψ) introduced above. This points of view avoids defining an infinite-dimensional Grassmann algebra. The equivalence relation among kernels, to be introduced momentarily, generalizes the relationships which hold between different ways of writing the coefficients of a given Grassmann polynomial. As mentioned above, in order to obtain bounds on the kernels of the effective potentials which are uniform in the scale label, we will need to group some of the Grassmann fields into discrete derivatives; we will mainly use the directional derivative ∂j φω,z := φω,z+êj − φω,z (note that this is the same 1090 G. Antinucci et al. Ann. Henri Poincaré convention used in Sect. 2.2.2). We consequently consider kernels which specify when and how this is done, and in particular define the equivalence relationship with this in mind. Let Λ∞ := Z2 , let B denote the set of nearest neighbor edges of Λ∞ , and let D := {D ∈ {0, 1, 2}2 : D1 ≤ 2}. Let M∞ = ∪n∈2N ({+, −} × D × Λ∞ )n be the set of field multilabels. for some n ∈ 2N, such that Di 1 ≤ 2. We can think of any Ψ = ((ω1 , D1 , z1 ), . . . , (ωn , Dn , zn )) ∈ M∞ as indexing a formal Grassmann monomial φ(Ψ) given by φ(Ψ) = ∂ D1 φω1 ,z1 · · · ∂ Dn φωn ,zn , (4.1.1) where, denoting Di = ((Di )1 , (Di )2 ) ∈ D, we let (Di )1 (Di )2 φωi ,zi , ∂2 ∂ Di φωi ,zi := ∂1 with ∂1 and ∂2 the (right) discrete derivatives introduced above. In the following, with some abuse of notation, any element Ψ ∈ M∞ of length |Ψ| = n will be denoted indistinctly by Ψ = ((ω1 , D1 , z1 ), . . . , (ωn , Dn , zn )) or Ψ = (ω, D, z), with the understanding that ω = (ω1 , . . . , ωn ), etc. We will call a function V : M∞ → R a kernel function, let Vn denote its restriction to field multilabels of length n, and let Vn,p be the restriction of Vn to field multilabels with D1 = p. Thinking of such a V as the coefficient function of a formal Grassmann polynomial  V (Ψ)φ(Ψ) (4.1.2) V(φ) = Ψ∈M∞ suggests an equivalence relationship corresponding to manipulations allowed by the anticommutativity of the Grassmann variables and by the definition of discrete derivative. More precisely, we say that V is equivalent to V ′ , and write V ∼ V ′ , if either: 1. V ′ is obtained from V by permuting the arguments and changing the sign according to the parity of the permutation; 2. V ′ is obtained from V by writing out the action of a derivative: that is, there exist n ∈ 2N, i ∈ {1, . . . , n} and j ∈ {1, 2} such that, letting − D+ i,j = (D1 , . . . , Di−1 , Di +êj , Di+1 , . . . , Dn ) and z i,j = (z1 , . . . , zi−1 , zi − êj , zi+1 , . . . , zn ), ⎧ ⎪ ⎨0 ′ − + Vn,p (ω , D , z ) = Vn (ω , D + i,j , z i,j ) − Vn (ω , D i,j , z ) ⎪ ⎩ + − Vn (ω , D , z )+Vn (ω , D i,j , z i,j )−Vn (ω , D + i,j , z ) if if if (Di )j = 2, (Di )j = 1, (Di )j = 0, (4.1.3) while Vm′ = Vm for all m ∈ 2N \ {n}; 3. V ′ is obtained from V by adding an arbitrary kernel V ∗ that is different from zero only for arguments with common repetition, that is, V ∗ (ω, D, z) = 0 unless there is some i = j such that (ωi , Di , zi ) = (ωj , Dj , zj ); Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1091 or V ′ is obtained from V by a countable sequence of such elementary operations and of convex combinations thereof. Moreover, we assume that the equivalence ′ relation ∼ is preserved by linear  i.e., if Vα ∼ Vα for all α in the  combinations, ′ countable index set I, then α∈I Vα ∼ α∈I Vα . We will call the equivalence classes generated by ∼ potentials and often specify them by formal sums like (4.1.2). Remark 4.1. The operation in item 2 can be thought of as a form of ‘integration by parts’. The kernels equivalent to zero, V ∼ 0, correspond to what are known as ‘null fields’ in the literature on conformal field theories. 4.2. Localization and Interpolation In this section we define the operators L∞ and R∞ acting on kernels indexed by field multilabels in M∞ and show several estimates related to R∞ . We recall that, given a kernel V , the symbol Vn,p denotes its restriction to field multilabels of length n, such that D1 = p. The operator L∞ . First of all, we let n (4.2.1) L∞ (Vn,p ) := 0, if 2 − − p < 0. 2 In the RG jargon, the combination 2 − n2 − p is the scaling dimension of Vn,p and will reappear below, for example in Lemma 4.7; in this sense, (4.2.1) says that the local part of the terms with negative scaling dimension (the irrelevant terms, in the RG jargon) is set equal to zero. There are only a few cases for which 2 − n2 − p ≥ 0, namely (n, p) = (2, 0), (2, 1), (4, 0). In these cases, the action of L∞ on Vn,p is non-trivial and $ which will be defined in terms of other basic operators, the first of which is L, $ n,p ) = (LV $ )n,p ≡ LV $ n,p , with is defined as: L(V ⎛ $ n,p (ω , D , (z1 , . . . , zn )) := ⎝ LV n  j=2 ⎞ δzj ,z1 ⎠  Vn,p (ω , D , (z1 , y2 , . . . , yn )). y2 ,...,yn ∈Λ∞ (4.2.2) If (n, p) = (2, 1), (4, 0), we let $ 2,1 ), L∞ (V2,1 ) := A(LV $ 4,0 ), L∞ V4,0 := A(LV (4.2.3) where A is the operator that antisymmetrizes with respect to permutations and symmetrizes with respect to reflections in the horizontal and vertical directions9 . A first important remark, related to the definitions (4.2.3), is that, if V2,1 is invariant under translations and under the action of A, then L∞ (V2,1 ) = c1 Fζ,∞ + c2 Fη,∞ , 9 The (4.2.4) reflection transformations in the infinite plane act on the Grassmann fields in the same way as (2.4.5), with the difference that the reflections θ1 (x1 , x2 ) and θ2 (x1 , x2 ) are replaced by their infinite-plane analogues, namely by θ̃1 (x1 , x2 ) = (−x1 , x2 ) and θ̃2 (x1 , x2 ) = (x1 , −x2 ), respectively. 1092 G. Antinucci et al. Ann. Henri Poincaré for some real numbers c1 , c2 and Fζ,∞ , Fη,∞ the A-invariant kernels associated with the potentials     φω,z d2 φ−ω,z , ωφω,z d1 φω,z , Fη,∞ (φ) := Fζ,∞ (φ) := ω=± z∈Λ∞ ω=± z∈Λ∞ with dj the symmetric discrete derivatives, acting on the Grassmann fields as dj φω,z = 12 (∂j φω,z + ∂j φω,z−êj ). A second and even more important remark $ 4,0 is is that, due to the fact that ω only assumes two values and that LV supported on z such that z1 = z2 = z3 = z4 , one has L∞ (V4,0 ) = 0, (4.2.5) a cancellation that will play an important role in the following. In order to define the action of L∞ on V2,0 , we want to obtain a kernel $ 2,0 , denoted by (RV $ )2,1 , which is supported function equivalent to V2,0 − LV on arguments with an additional derivative. As we will see, the definition of $ will also play a central role in the definition of the operator R∞ below. We R rewrite  $ 2,0 (ω, 0, z)]φ(ω, 0, z) [V2,0 (ω, 0, z) − LV z ∈Λ2∞ =  z ∈Λ2∞ =  z ∈Λ2∞ V2,0 (ω, 0, z)[φ(ω, 0, z) − φ(ω, 0, (z1 , z1 ))] V2,0 (ω, 0, z)φω1 ,z1 (φω2 ,z2 − φω2 ,z1 ). (4.2.6) We now intend to write the difference φω1 ,z1 (φω2 ,z2 − φω2 ,z1 ) as a sum of terms ′ of the form φω1 ,z1 ∂ D φω2 ,y , with D′ 1 = 1, over the sites y on a path from z1 to z2 . To do this we must specify which path is to be used. For each z, z ′ ∈ Λ∞ , let γ(z, z ′ ) = (z, z1 , z2 , . . . , zn , z ′ ) be the shortest path obtained by going first horizontally and then vertically from z to z ′ . Note that γ is covariant under the symmetries of the model on the infinite plane, i.e., Sγ(z, z ′ ) = γ(Sz, Sz ′ ) (4.2.7) where S : Λ∞ → Λ∞ is some composition of translations and reflections parallel to the coordinate axes. Given z, z ′ two distinct sites in Λ∞ , let INT(z, z ′ ) be the set of (σ, (D1 , D2 ), (y1 , y2 )) ≡ (σ, D, y) ∈ {±} × {0, ê1 , ê2 }2 × Λ2∞ such that: (1) y1 = z, (2) D1 = 0, (3) y2 , y2 + D2 ∈ γ(z, z ′ ), (4) σ = + if y2 precedes y2 + D2 in the sequence defining γ(z, z ′ ), and σ = − otherwise. In terms of this definition, one can easily check that (4.2.6) can be rewritten as   V2,0 (ω, 0, z) σφ(ω, D, y) (4.2.6) = z ∈Λ2∞ (σ,D ,y )∈INT(z ) (4.2.8) (1) ≡   y ∈Λ2∞ D $ )2,1 (ω, D, y)φ(ω, D, y), (RV Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1093 where, if z = (z1 , z1 ), the sum over (σ, D, y) in the first line should be interpreted as being equal to zero (in this case, we let INT(z1 , z1 ) be the empty set). In going from the first to the second line, we exchanged the order of (p) summations over z and y; moreover, D denotes the sum over the pairs D = (D1 , D2 ) such that D1 = p, and  $ )2,1 (ω, D, y) := (RV σV2,0 (ω, 0, z). (4.2.9) σ,z : (σ,D ,y )∈INT(z ) $ 2,0 . We $ )2,1 ∼ V2,0 − LV From the previous manipulations, it is clear that (RV are finally ready to define the action of L∞ on V2,0 : $ 2,0 + L( $ RV $ )2,1 ). L∞ (V2,0 ) := A(LV (4.2.10) L∞ (V2,0 ) = c0 Fν,∞ + c1 Fζ,∞ + c2 Fη,∞ , (4.2.11) Note that, if V2,0 is invariant under translations and under the action of A, then for some real numbers c0 , c1 , c2 , and Fν,∞ the A-invariant kernel associated with the potential 1   Fν,∞ (φ) := ωφω,z φ−ω,z , (4.2.12) 2 ω=± z∈Λ∞ while we recall that Fζ,∞ , Fη,∞ were defined right after (4.2.4). Summarizing, in view of Eq. (4.2.5), ⎧ $ ⎪ if (n, p) = (2, 0), ⎨A (LV2,0 ) $ $ $ (L∞ V )n,p = A(LV2,1 + L(RV )2,1 ) if (n, p) = (2, 1), (4.2.13) ⎪ ⎩ 0 otherwise. The operator R∞ . We now want to define an operator R∞ such that R∞ V ∼ V − L∞ V for kernels V that are invariant under translations and under the action of A. First of all, we let R∞ (Vn,p ) = (R∞ V )n,p := Vn,p , or n = 2 and p ≥ 3. if: n ≥ 6, or n = 4 and p ≥ 2, (4.2.14) Moreover, we let (R∞ V )2,0 = (R∞ V )2,1 = (R∞ V )4,0 := 0. (4.2.15) The only remaining cases are (n, p) = (2, 2), (4, 1). For these values of (n, p), (R∞ V )n,p is defined in terms of an interpolation generalizing the definition $ )2,1 in (4.2.9). As a preparation for the definition, we first introduce of (RV $ (RV )n,p for (n, p) = (2, 2), (4, 1). For this purpose, we start from the analogues of (4.2.6), (4.2.8) in the case that (2, 0) is replaced by (n, p) = (2, 1), (4, 0): for such values of (n, p) we write 1094 G. Antinucci et al.  z ∈Λn ∞ = $ n,p (ω, D, z)]φ(ω, D, z) [Vn,p (ω, D, z) − LV  z ∈Λn ∞  = Ann. Henri Poincaré Vn,p (ω, D, z)[φ(ω, D, z) − φ(ω, D, (z1 , z1 , . . . , z1 ))] Vn,p (ω, D, z) z ∈Λn ∞  σφ(ω, D + D ′ , y). (4.2.16) (σ,D ′ ,y )∈INT(z ) In the last expression, if n = 2, then INT(z) is the same defined after (4.2.7); if n = 4, then INT(z) is the set of (σ, (D1 , . . . , D4 ), (y1 , . . . , y4 )) ≡ (σ, D, y) ∈ {±} × {0, ê1 , ê2 }4 × Λ4∞ such that: either y1 = y2 = y3 = z1 , D1 = D2 = D3 = 0, and (σ, (0, D4 ), (z1 , y4 )) ∈ INT(z1 , z4 ); or y1 = y2 = z1 , y4 = z4 , D1 = D2 = D4 = 0, and (σ, (0, D3 ), (z1 , y3 )) ∈ INT(z1 , z3 ); or y1 = z1 , y3 = z3 , y4 = z4 , D1 = D3 = D4 = 0, and (σ, (0, D2 ), (z1 , y2 )) ∈ INT(z1 , z2 ). By summing (4.2.16) over D and exchanging the order of summations over z and y, we find (p)   z ∈Λn ∞ = Vn,p (ω, D, z) with σφ(ω, D + D ′ , y) (σ,D ′ ,y )∈INT(z ) D  (p+1)  y ∈Λn ∞  D (4.2.17) $ )n,p+1 (ω, D, y)φ(ω, D, y). (RV $ )n,p+1 (ω, D, y) := (RV  ′ σ,z ,D : (σ,D ′ ,y )∈INT(z ) σVn,p (ω, D − D ′ , z). (4.2.18) We are now ready to define: $ )2,2 + (R( $ RV $ ))2,2 ), (R∞ V )2,2 := A(V2,2 + (RV $ )4,1 ). (R∞ V )4,1 := A(V4,1 + (RV (4.2.19) Summarizing, (R∞ V )n,p ⎧ 0 ⎪ ⎪ ⎪ ⎨A(V + (RV $ )2,2 + (R( $ RV $ ))2,2 ) 2,2 = $ ⎪A(V4,1 + (RV )4,1 ) ⎪ ⎪ ⎩ Vn,p if (n, p) = (2, 0), (2, 1), (4, 0), if (n, p) = (2, 2), if (n, p) = (4, 1), otherwise (4.2.20) From the previous manipulations and definitions, it is clear that, if V is invariant under translations and under the action of A, then V − L∞ V ∼ R∞ V . For later use, given D = (D1 , . . . , Dn ) with D1 = p, we let R∞ V D be the restriction of (RV )n,p to that specific choice of derivative label. Remark 4.2. From the definitions of L∞ and R∞ , it also follows that, if V is invariant under translations and under the action of A, then L∞ (L∞ V ) = L∞ V and R∞ (L∞ V ) = 0, two properties that will play a role in the following. Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1095 Norm bounds. Let us conclude this section by a couple of technical estimates, which relate a suitable weighted norm of R∞ V to that of V , and will be useful in the following. Suppose that V is translationally invariant. Let, for any κ ≥ 0,  (p) eκδ(z ) sup |Vn,p (ω, D, z)|, (4.2.21) Vn,p (κ) := sup ω D z ∈Λn ∞: z1 fixed where the label (p) on the sup over D indicates the constraint that D1 = p, and δ(z) is the tree distance of z. With these definitions, Lemma 4.3. For any positive ǫ, $ )n,p (κ) ≤ (n − 1)ǫ−1 Vn,p−1 (κ+ǫ) if (RV (n, p) = (2, 2), (4, 1), (4.2.22) $ RV $ ))2,2 (κ) ≤ ǫ−2 V2,0 (κ+2ǫ) . (R( (4.2.23) As a consequence, noting that AVn,p (κ) ≤ Vn,p (κ) and recalling the definitions (4.2.20), we find that (R∞ V )2,2 (κ) ≤ V2,2 (κ) + ǫ−1 V2,1 (κ+ǫ) + ǫ−2 V2,0 (κ+2ǫ) , −1 (R∞ V )4,1 (κ) ≤ V4,1 (κ) + 3ǫ V4,0 (κ+ǫ) . (4.2.24) (4.2.25) In the following, we will use bounds of this kind in order to evaluate the size of the renormalized part of the effective potential on scale h. In such a case, both κ and ǫ will be chosen of the order 2h . Proof. In order to prove (4.2.22), note that it follows directly from the defini$ that tion of R   (p) $ )n,p (κ) ≤ sup (RV eκδ(z ) |Vn,p−1 (ω, D − D ′ , y)|. sup ω z ∈Λn ∞: z1 fixed D σ,y ,D ′ : (σ,D ′ ,z )∈INT(y ) (4.2.26) If we now exchange the order of summations over z and y, we find   (p−1) $ )n,p (κ) ≤ sup (RV eκδ(z ) sup |Vn,p−1 (ω, D, y)|. ω y ∈Λn σ,z ,D ′ : ∞: y1 fixed (σ,D ′ ,z )∈INT(y ) D (4.2.27) Now note that δ(z) ≤ δ(y) and that | INT(y)| ≤ (n − 1)δ(y), so that  (p−1) $ )n,p (κ) ≤ (n − 1) sup (RV eκδ(y ) δ(y) sup |Vn,p−1 (ω, D, y)| ω y ∈Λn ∞: y1 fixed n−1 ≤ Vn,p−1 κ+ǫ , ǫ D (4.2.28) 1096 G. Antinucci et al. Ann. Henri Poincaré where in the last step we used the fact that δ ≤ eǫδ /ǫ, for any ǫ > 0. A two-step iteration of the bound (4.2.22) proves (4.2.23).  Similar estimates are valid for more general values of (n, p), but Lemma 4.3 includes all the cases which are relevant to the present work. The running coupling constants. At each scale h ≤ 0 we represent the infinite volume effective potential, as arising from the iterative application of (4.3) in the infinite volume limit, in the form (4.1.2), namely:  (h) (h) V∞ V∞ (Ψ)φ(Ψ). (4.2.29) (φ) = Ψ∈M∞ (h−1) (h) For each h ≤ 0, in order to compute V∞ from V∞ via the infinite volume (h) (h) (h) limit of (4.3), we decompose V∞ ∼ L∞ V∞ + R∞ V∞ . Note that the kernel (h) L∞ V∞ , in light of (4.2.4), (4.2.5) and (4.2.4), takes the form (h) L∞ V∞ = 2h νh Fν,∞ + ζh Fζ,∞ + ηh Fη,∞ =: υh · F∞ , (4.2.30) for three real constants νh , ζh , ηh , called the running coupling constants at by the fact that the Fν,∞ = scale h. The factor 2h in front of νh is motivated  (Fν,∞ )2,0 has scaling dimension 2 − n2 − p(n,p)=(2,0) = 1, see (4.2.1), see also (4.3.10) below. 4.3. Trees and Tree Expansions In this section, we describe the expansion for the kernels of the effective potentials, as it arises from the iterative application of Eq. (4.3). As anticipated above, it is convenient to graphically represent the result of the expansion in terms of GN trees. At the first step, recalling (3.23) and denoting by Nc (Ψ) and Nm (Ψ) the kernels of Nc (φ) and Nm (ξ), respectively, we reorganize the (0) expression for V∞ obtained by taking the infinite volume of Eq. (4.4) to obtain, for any Ψ = (ω, D, z) ∈ M∞ such that D = 0 (which we identify with the corresponding element (ω, z) of ∪n∈2N ({+, −} × Λ∞ )n ), (0) int V∞ (Ψ) = Nc (Ψ) + Z −|Ψ|/2 V∞ (Ψ) + ⎛  (1) int Z −|Ψ1 |/2 V∞ (Ψ1 )G∅,∞ (Ψ1 \ Ψ) Ψ1 ∈M1,∞ : Ψ1 ⊃Ψ ⎞ (Ψ) s ∞     1 −|Ψ |/2 int j ⎝ Nm (Ψj ) + Z + V∞ (Ψj ) ⎠ α(Ψ; Ψ1 , . . . , Ψs ) s! Ψ ,...,Ψ ∈M s=2 j=1 1 s 1,∞  (1) × GT ,∞ (Ψ̄1 , . . . , Ψ̄s ), (4.3.1) T ∈S(Ψ̄1 ,...,Ψ̄s ) int is the infinite volume limit of the kernel of V int (φ, ξ) := V int (φ, ξ, 0), where V∞ (1) and we recall that in the first (resp. last) line, the factor GT,∞ (Ψ1 \ Ψ) (resp. (1) GT,∞ (Ψ̄1 , . . . , Ψ̄s )) is different from zero only if Ψ1 \ Ψ is an element (resp. Ψ̄1 , . . . , Ψ̄s are elements) of ∪n∈2N ({+i, −i} × Λ∞ )n . For the definition of Ψ̄j , see the first item in the list after Eq. (4.4). Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1097 Figure 1. Graphical interpretation of (4.3.1) Similarly, at the following steps, for any h ≤ 0 and any Ψ ∈ M∞ , we (h−1) can write V∞ (Ψ), as computed via the infinite volume analogue of (4.3), as follows: (h−1) V∞ (h) (Ψ) ∼ υh · F∞ (Ψ) + R∞ V∞ (Ψ) +  (h) (h) R∞ V∞ (Ψ1 )G∅,∞ (Ψ1 \ Ψ) Ψ1 ∈M1,∞ : Ψ1 ⊃Ψ ⎞ ⎛ (Ψ) ∞ s     1 (h) ⎝ υh · F∞ (Ψj )+R∞ V∞ (Ψj ) ⎠ α(Ψ; Ψ1 , . . . , Ψs ) + s! Ψ ,...,Ψ ∈M s=2 j=1 1 s 1,∞  (h) GT ,∞ (Ψ̄1 , . . . , Ψ̄s ), (4.3.2) × T ∈S(Ψ̄1 ,...,Ψ̄s ) (h) where GT,∞ (Ψ̄1 , . . . , Ψ̄s ) is the infinite volume analogue of the function defined in (4.5), differing from it for an important feature (besides the ‘obvious’ one (h) (h) that GT,∞ is defined in terms of the infinite plane propagators gℓ,∞ rather than those on the cylinder): since now the field multilabels Ψi have the form (ω i , D i , z i ), with D i different from 0, in general, the infinite plane propagators (h) (h) gℓ,∞ , with ℓ = ((ωi , Di , zi ), (ωj , Dj , zj )), entering the definition of GT,∞ should D (h) now be interpreted as ∂zDi i ∂zj j gωi ωj (zi , zj ). We graphically interpret (4.3.1) as in Fig. 1. On the right-hand side of the first line, we have drawn a series of diagrams consisting of a root at scale 1, which we will usually denote v0 , connected to s other vertices (which we will call endpoints) at scale 2 which are of two different types: , which we call counterterm endpoints and which represent Nc or Nm , and , called interaction int endpoints and representing V∞ . In the first two terms (in which there is no (1) (1) factor GT,∞ , because it is ‘trivial’, i.e., it equals G∅,∞ (∅) = 1) the root is drawn simply as the end of a line segment (we will say it is undotted); in the other terms (including all those with s > 1) we draw a dot to indicate the (1) presence of a non-trivial GT,∞ and α factors and additional sums. (0) (0) (0) In order to iterate the scheme, we decompose V∞ as V∞ ∼ L∞ V∞ + (0) (0) R∞ V∞ and graphically represent L∞ V∞ = υ0 · F∞ by a counterterm vertex (0) at scale 1, and R∞ V∞ by a big vertex at scale 1, as indicated in the second line of Fig. 1. Next, using the conventions of Fig. 1, we graphically (−1) represent V∞ , computed by (4.3.2) with h = 0, as described in Fig. 2. 1098 G. Antinucci et al. Ann. Henri Poincaré Figure 2. Graphical interpretation of (4.3.2) with h = 0 In passing from the first to the second line of Fig. 2 we expanded the big (0) vertex on scale 1, which represents R∞ V∞ , by using the first line of Fig. 1, with an additional label R∞ on the vertices on scale 1, to represent the action of R∞ . The graphical equations in Figs. 1 and 2 are the analogues of the graphical equations in [15, Figures 6-7], which contains a more detailed discussion of some aspects of this construction. By iterating the same kind of graphical equations on lower scales, expanding the big vertices until we are left with (h) endpoints all of type or , we find that V∞ can be graphically expanded in terms of trees of the kind depicted in Fig. 3, with the understanding that in principle there should be a label R∞ at all the intersections of the branches with the vertical lines, with the sole exception of v0 ; however, by convention, in order not to overwhelm the figures, we prefer not to indicate these labels (h) explicitly. We call such trees ‘GN trees’ and denote by T∞ , with h ≤ 0, the set of GN trees with root v0 on scale h + 1. We call ‘vertices’ of a GN tree the root v0 , all its dotted nodes, and its endpoints. We introduce some conventions and observations about the set of GN trees: • The root v0 is the unique leftmost vertex of the tree. Its degree (number of incident edges) must be at least 1, i.e., v0 cannot be an endpoint. It may or may not be dotted; in order for v0 not to be dotted, its degree must be 1. • Vertices, other than the root, with exactly one successor, are called ‘trivial’. • Interaction endpoints can only be on scale 2. Counterterm endpoints can be on all scales ≤ 2; if such an endpoint is on a scale h < 2, then it must be connected to a non-trivial vertex on scale h − 1. [The reason is the following: if this were not case, then there would be an R∞ operator Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1099 v0 h+1 2 (h) Figure 3. Example of a tree in T∞ . As explained in the text, one should imagine that a label R, indicating an action of the R operator, is present at all the intersections of the branches with the vertical lines, with the only exception of v0 . In order not to overwhelm the figures, these labels are left implicit acting on the value of the endpoint, but this would annihilate it, because (h) a endpoint on scale h < 2 corresponds to L∞ V∞ , and the definitions (h) of L∞ , R∞ are such that R∞ (L∞ V∞ ) ∼ 0, see Remark 4.2.] (h) (h) In terms of these trees, we shall write the expansion for V∞ = V∞ [υ], thought of as a function of υ := {(νh , ζh , ηh )}h≤0 , as  (h) V∞ W∞ [υ; τ ]. (4.3.3) [υ] ∼ (h) τ ∈T∞ In order to write W∞ [υ; τ ] more explicitly, we need to specify some additional (h) notations and conventions about GN trees. Let τ ∈ T∞ := ∪h≤0 T∞ and v0 = v0 (τ ) be its root. Then: • We let V (τ ) be the set of vertices, Ve (τ ) ⊂ V (τ ) the set of endpoints, and V0 (τ ) := V (τ ) \ Ve (τ ). We also let V ′ (τ ) := V (τ ) \ {v0 } and V0′ (τ ) := V0 (τ ) \ {v0 }. • Given v ∈ V (τ ), we let hv be its scale. 1100 G. Antinucci et al. Ann. Henri Poincaré • v ≥ w or ‘v is a successor of w’ means that the (unique) path from v to v0 passes through w. Obviously, v > w means that v is a successor of w and v = w. • ‘v is an immediate successor of w’, denoted v ⊲ w, means that v ≥ w, v = w, and v and w are directly connected. For any v ∈ τ , Sv is the set of w ∈ τ such that w ⊲ v. • For any v > v0 , we denote by v ′ the unique vertex such that v ⊲ v ′ . (h −1) • Subtrees: for each v ∈ V0 (τ ), let τv ∈ T∞ v denote the subtree consisting of the vertices with w ≥ v. Next, we need to attach labels to their vertices, in order to distinguish the various contributions to the kernels arising from the different choices of the sets Ψi , etc., in (4.3.1), (4.3.2), also keeping track of the order in which they appear. In particular, with each v ∈ V (τ ) we associate a set Pv of field labels, sometimes called the set of external fields, whose elements carry two informations: their position within an ordered list which they belong to, and their ω index; more precisely, the family P = {Pv }v∈V (τ ) is characterized by the following properties, which correspond to properties of the iteration of the kernel: • |Pv | is always even and positive. If v is a endpoint, then |Pv | = 2. • If v is an endpoint of τ , then Pv has the form {(j, 1, ω1 ), . . . , (j, 2n, ω2n )}, where j is the position of v in the ordered list of endpoints, and ωl ∈ {+, −, i, −i}, if hv = 2, while ωl ∈ {+, −}, if hv < 2. Given f = (j, l, ωl ), we let o(f ) = (j, l) and ω(f ) := . ωl . • If v is not an endpoint, Pv ⊂ w∈Sv Pw . . • If v ∈ V0 (τ ), we let Qv := w∈Sv Pw \ Pv be the set of contracted fields. If v is dotted, then we require |Qv | ≥ 2 and |Qv | ≥ 2(|Sv | − 1); and conversely Qv is empty if and only if v = v0 and 0 . v0 is /not dotted. • If hv = 1 and v is not an endpoint, then Qv = w∈Sv f ∈ Pw  ω(f ) ∈ {+i, −i} (all and only massive fields are integrated on scale 1). For τ ∈ T∞ , we denote by P(τ ) be the set of allowed P = {Pv }v∈V (τ ) . We also denote by ω v the tuple of components ω(f ), with f ∈ Pv , and by ω v Q the restriction of ω v to any subset Q ⊆ Pv . Note that the definitions imply that for v, w ∈ τ such that neither v ≥ w or v ≤ w (for example when v ′ = w′ but v = w), Pv and Pw are disjoint, as are Qv and Qw . Next, given P ∈ P(τ ), for all v ∈ V0 (τ ) we define sets Tv , 0 /  Tv = (f1 , f2 ) , . . . , f2|Sv |−3 , f2|Sv |−2 ⊂ Q2v , called spanning trees associated with v, characterized by the following properties: if w(f ) denote the (unique) w ∈ Sv for which f ∈ Pw , ′ then (f, f ′ ) ∈ Tv /⇒ w(f ) = w(f ′ ) and / 00 o(f ) < o(f ); moreover, {w(f1 ), w(f2 )} , . . . , w(f2|Sv |−3 ), w(f2|Sv |−2 ) is the edge set of a tree with vertex set Sv . We denote by S(τ, P ) the set of allowed T = {Tv }v∈V0 (τ ) . Finally, for each v ∈ V (τ ), we denote by Dv a map Dv : Pv → D = {D ∈ {0, 1, 2}2 : D1 ≤ 2}; the reader should think that a derivative operator ∂ Dv (f ) acts on the field labeled f . We denote by D(τ, P ) the set of families Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1101 of maps D = {Dv }v∈V (τ ) . We also denote by D v the tuple of components Dv (f ), with f ∈ Pv , and by D v Q the restriction of D v to any subset Q ⊆ Pv . Additionally, if a map z : Pv → Λ∞ is assigned, we denote by z v the tuple of components z(f ), with f ∈ Pv , and by Ψv = Ψ(Pv ) := (ω v , D v , z v ) the field multilabel associated with ω v , D v , z v ; moreover, if v ∈ V0 (τ ) and also the maps z : Pw → Λ∞ , for all w ∈ Sv , are  assigned, for each w ∈ Sv we denote by Ψ̄w = Ψ(Pw \ Pv ) = (ω w P \P , D w P \P , z w P \P ) the restriction of Ψw w v w v w v  to Pw \ Pv (here z w  is the restriction of z w to the subset Q ⊂ Pw ). Q In terms of these definitions, we write W∞ [υ; τ ] in the right side of (4.3.3) as W∞ [υ; τ ] =    W∞ [υ; τ, P , T , D], (4.3.4) P ∈P(τ ) T ∈S(τ,P ) D∈D(τ,P ) where W∞ [υ; τ, P , T , D] is the translationally  invariant kernel inductively 1 defined as follows: letting D ′v0 := v∈Sv D v P for hv0 < 1 and D ′v0 := 0 v0 0 for hv0 = 1,  α  v0 W∞ [υ; τ, P , T , D](ω 0 , D 0 , z 0 ) = 1(ω 0 = ω v0 )1 D 0 = D v0 = D ′v0 |Sv0 |!   (h ) (4.3.5) ) G v0 (Ψ̄ , . . . , Ψ̄ K (Ψ ), × z:Pv0 ∪Qv0 →Λ∞ z 0 =z v0 Tv0 ,∞ v1 v sv v,∞ 0 v v∈Sv0 where αv0 = α(Ψv0 ; Ψv1 , . . . , Ψvsv ), cf. (4.3.1), and we recall that, if |Sv0 | = 1, 0 (h ) v0 then Tv0 = ∅. In this case, if Ψv1 = Ψv0 , then G∅,∞ (∅) should be interpreted as being equal to 1; this latter case is the one in which, graphically, v0 is not dotted. In the second line of (4.3.5), if hv0 = 1, ⎧ ⎪ if v is of type and v0 is undotted, ⎨Nc (Ψ) Kv,∞ (Ψv ) = Nm (Ψv ) if v is of type and v0 is dotted, ⎪ ⎩ −|Ψv |/2 int Z V∞ (Ψv ) if v is of type , (4.3.6) while, if hv0 < 1, Kv,∞ (Ψv ) := ⎧ ⎪ ⎪ ⎪υhv0 · F∞ (Ψv ) ⎪ ⎨R N (Ψ ) ∞ c v −|Ψv |/2 ⎪ Z ⎪ ⎪ ⎪ ⎩W int R∞ V∞ (Ψv ) ∞ [υ; τv , P v , T v , D v ](Ψv ) if v ∈ Ve (τ ) is of type and hv = hv0 +1 if v ∈ Ve (τ ) is of type and hv = 2 if v ∈ Ve (τ ) is of type if v ∈ V0 (τ ), (4.3.7) where, in the last line of (4.3.7), letting P v (resp. T v , resp. Dv ) be the ′ ′ restriction of P (resp. T , resp. D) to the subtree τv , and 1 Dv := {Dv } ∪ ′ ′ {Dw }w∈V (τ ):w>v0 (here Dv is the map such that D v := w∈Sv D w P ), we v denoted   (4.3.8) W ∞ [υ; τv , P v , T v , Dv ] := R∞ W∞ [υ; τv , P v , T v , D′v ] , Dv 1102 G. Antinucci et al. Ann. Henri Poincaré  and we recall that the definition of R∞ V D was given a few lines after (4.2.20). (h) (h) The inductive proof that V∞ = V∞ [υ], as iteratively computed by (4.3.1)– (4.3.2), is equivalent to τ ∈T∞(h) W∞ [υ; τ ], with W∞ [υ; τ ] as in (4.3.4), (4.3.5), is straightforward and left to the reader. Remark 4.4. Given τ ∈ T∞ and P ∈ P(τ ), we say that D is ‘allowed’ if W∞ [υ; τ, P , T , D] ∼ 0. With some abuse of notation, from now on we will redefine D(τ, P ) to be the set of allowed D for a given τ and P . Of course, such a redefinition has no impact on the validity of (4.3.4). If D is allowed, then it must satisfy a number of constraints. In particular, if v ∈ V0 (τ ), w ∈ Ve (τ ) , then Dv (f) ≥ Dw (f ). Moreover, letting, for any v ∈ V0 (τ ), and f ∈ Pv ∩ Pw Rv := D v 1 − w∈Sv D w P 1 ≡ D v 1 − D ′v 1 , one has v ⎧ 2 ′2 2D v 2 = 0 ⎪ 2, |P | = 2 and v ⎨ 2 21 2 2 Rv = 1, |Pv | = 2 and 2D ′v 21 = 1 or |Pv | = 4 and 2D ′v 21 = 0 ⎪ ⎩ 0, otherwise, (4.3.9) with the exception of v0 , for which Rv0 ≡ 0 (in other words D v0 = D ′v0 , see (4.3.5)). Finally, the combination |Pv | − D v 1 , (4.3.10) 2 known as the scaling dimension of v, is ≤ −1 for all v ∈ V0′ (τ ), and for all v ∈ Ve (τ ) such that hv = 2 and hv′ < 1. As we shall see below, see in particular the statement of Lemma 4.7, the fact that d(Pv , D v ) ≤ −1 for all such vertices guarantees that the expansion in GN trees is convergent uniformly in hv0 . d(Pv , D v ) := 2 − Remark 4.5. With the other arguments fixed, the number of choices of D for which W∞ [υ; τ, P , T , D] does not vanish is no more than 10|V (τ )| : there is a choice of at most 10 possible values for each endpoint10 , and then the other values are fixed except for a choice of up to 10 possibilities each time that R∞ acts non-trivially on a vertex v ∈ V0′ (τ ), i.e., each time that, for such a vertex, Rv > 0, see (4.3.9). 4.4. Bounds on the Kernels of the Full Plane Effective Potentials In this subsection we show that the norm of the kernels W∞ [υ; τ, P , T , D] is summable over τ, P , T , D, provided that the elements of the sequence υ are bounded and sufficiently small. We measure the size of W∞ [υ; τ, P , T , D] in terms of the weighted norm (4.2.21), with κ = c20 2hv0 , where hv0 is the scale of 10 The value 10 bounds the number of different terms that the operator R∞ produces when it acts non-trivially on an interaction endpoints. In fact, the cases in which R∞ acts nontrivially are those listed in the right side of (4.2.20) with (n, p) = (2, 2), (4, 1). If (n, p) = (2, 2), the number of possible values taken by Dv is 10 (one derivative in direction i ∈ {1, 2} on the first Grassmann field and one derivative in direction j ∈ {1, 2} on the second Grassmann field, etc.); if (n, p) = (4, 1), the number of possible values taken by Dv is 8, which is smaller than 10 (one derivative in direction i ∈ {1, 2} on the k-the Grassmann field, with k ∈ {1, 2, 3, 4}). Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1103 the root of τ and c0 is the minimum between the constant c in Proposition 2.3 and half the constant c in (3.14), (3.15), (3.20), (3.21). With some abuse of notation, we let W∞ [υ; τ, P , T , D]hv0 := W∞ [υ; τ, P , T , D]( c0 2hv0 ) . (4.4.1) 2 The first, basic, bound on the kernels W∞ [υ; τ, P , T , D] is provided by the following proposition. We recall that we assumed once and for all that Z ∈ U , with U = {z ∈ R : |z −1| ≤ 1/2}, and that t∗1 , t1 , t2 ∈ K ′ , with K ′ the compact set defined before the statement of Theorem 1.1. Proposition 4.6. Let W∞ [υ; τ, P , T , D] be inductively defined as in (4.3.5). There exist C, κ, λ0 > 0 such that, for any τ ∈ T∞ , P ∈ P(τ ), T ∈ S(τ, P ), D ∈ D(τ, P ), and |λ| ≤ λ0 , W∞ [υ; τ, P , T , D]hv0 ⎛   ≤ C v∈Ve (τ ) |Pv | ⎝ 1 2( 2 |Qv |+  w∈Sv D w |Qv 1 −Rv +2−2|Sv |)hv |Sv |! v∈V0 (τ ) ×  v∈Ve (τ ) 1 2(hv −1)(2− 2 |Pv |−D v 1 ) ǫhv −1 |λ|max{1,κ|Pv |} if v is of type if v is of type ⎞ ⎠ (4.4.2) where ǫh := max{|νh |, |ζh |, |ηh |} if h ≤ 0 and ǫ1 := max {Nc 2 , Nm 2 }. Proof. Let us first consider the case hv0 = 1, in which case, using (4.3.5) and (4.3.6), we find  1 W∞ [υ; τ, P , T , 0]hv0 ≤ |Sv0 |! z:∪v∈Sv Pv →Λ∞ : 0 z(f1 ) fixed   (1) ×ec0 δ(z v0 ) GTv ,∞ (Ψ̄v1 , . . . , Ψ̄vsv ) · 0 0 ⎧ ⎪ if v is of type and v0 is undotted,  ⎨|Nc (Ψv )| · and v0 is dotted, |Nm (Ψv )| if v is of type ⎪ v∈Sv0 ⎩|Z|−|Ψv |/2 |V int (Ψ )| if v is of type , v ∞ (4.4.3) (1) where f1 is the first element of Pv0 . By using the definition of GT,∞ and the property (PfM )2 = det M , valid for any antisymmetrix matrix M , we find  3  (1)   (1) (1)  G |g | sup | det G (Q1 , . . . , Qs ) ≤ (t)|, (4.4.4) T,∞ ℓ,∞ ℓ∈T t Q1 ,...,Qs ,T,∞ (1) so that, thanks to items 1,3,4 of Proposition 2.3 (which apply to g∞ by Remark 2.6), and to the Gram-Hadamard inequality [15, Appendix A.3], which allows one to bound the determinant of any matrix M with elements 1104 G. Antinucci et al.  Mi,j = (γi , γ̃j ) as | det M | ≤ |γi | |γ̃i |, ⎛    (1) s−1 ⎝  G T,∞ (Q1 , . . . , Qs ) ≤ C Ann. Henri Poincaré i e−2c0 z(f )−z(f (f,f ′ )∈T ⎛ ×⎝  f ∈∪i Qi \T  ≤ (C ′ ) If we now note that δ(z 0 ) ≤  (f,f ′ )∈Tv0 i |Qi | ⎛ ⎝ (1) |γω(f ),0,z(f ) |  · ′ ⎞ )1 ⎠ e−2c0 z(f )−z(f (f,f ′ )∈T z(f ) − z(f ′ )1 +  ⎞1/2 (1) |γ̃ω(f ),0,z(f ) |⎠ ′ ⎞ )1 ⎠ . δ(z v ), (4.4.5) (4.4.6) v∈Sv0 and plug Eq. (4.4.5) into (4.4.3), we find W∞ [υ; τ, P , T , 0]hv0 ≤ C  ⎧ ⎨(Nc )2 (c0 )  ⎪ × (Nm )2 (c0 ) ⎪ v∈Sv0 ⎩(V int ) ∞ |Pv | (c0 ) v∈Sv 0 |Pv | |Sv0 |!  −c0 z1 e z∈Λ∞ |Sv0 |−1 if v is of type and v0 is undotted, if v is of type and v0 is undotted, if v is of type , (4.4.7) which immediately implies the desired bound for hv0 = 1, because of the (1) definition of ǫ1 and (V∞ )|Pv | (c0 ) ≤ C |Pv | |λ|max{1,κ|Pv |} , see (3.15) and recall the definition of c0 at the beginning of this subsection. Next, we consider the case hv0 < 1, in which case Kv,∞ is defined by (4.3.7). We proceed similarly: we start from (4.3.5) and use the analogue of (4.4.4)–(4.4.5), namely      (hv ) (hv0 ) G 0 (Q1 , . . . , Qs ) ≤ | |g T,∞ ℓ,∞ (f,f ′ )∈T ⎛ ⎝  f ∈∪i Qi \T (h ) (h ⎞1/2 ) |γω(fv0),D(f ),z(f ) | · |γ̃ω(fv0),D(f ),z(f ) |⎠ 1 ≤ (C2hv0 ) 2  i |Qi | hv0 2  f ∈∪i Qi D(f )1 ⎛ ⎝  (f,f ′ )∈T e−c0 2 hv 0 z(f )−z(f ′ ) 1 ⎞ ⎠, (4.4.8) where in the second inequality we used again the bounds in items 1,4 of Prop.2.3. Using (4.4.8) and, again, (4.4.6), we obtain the analogue of (4.4.7), W∞ [υ; τ, P , T , D]hv 0 Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry ≤ · 1 |Sv0 |!  v∈Sv 0 (C2 hv 0 ) |Qv | 0 2 2 hv 0  v∈Sv 0 D v |Q v ⎧ (υhv · F∞ )2, D v 1 hv ⎪ ⎪ 0 0 ⎪ ⎪ ⎨(R N ) ∞ c 2, D v 1 hv 0 −|Ψv |/2 int ⎪ (R∞ V∞ )|Pv |, D v ⎪|Z| ⎪ ⎪ ⎩ (W ∞ [υ; τv , P v , T v , D v ])|Pv |, 0 1 ·  e c h − 0 2 v0 2 z∈Λ∞ 1 hv 0 D v 1 hv0 z 1 |Sv 1105 0 |−1 · if v ∈ Ve (τ ) is of type and hv = hv0 + 1 if v ∈ Ve (τ ) is of type and hv = hv0 + 1 if v ∈ Ve (τ ) is of type if v ∈ V0 (τ ), (4.4.9) The terms in the second line can be bounded as follows: • If v ∈ Ve (τ ) is of type and hv = hv0 + 1, (υhv0 · F∞ )2,D v 1 hv0 ≤ C 2hv0 |νhv0 | if D v 1 = 0 max{|ζhv0 |, |ηhv0 |} if D v 1 = 1 ≤ C ′ 2(hv −1)(2− |Pv | 2 −D v 1 ) ǫhv −1 (4.4.10) where we recall that ǫh = max{|νh |, |ζh |, |ηh |} and, in passing from the first to the second line, we used the fact that |Pv | = 2 and hv0 = hv − 1, |Pv | so that 2(hv −1)(2− 2 −D v 1 ) is equal to 2hv0 , if D v 1 = 0, and is equal to 1, if D v 1 = 1. and hv = hv0 + 1, using Lemma 4.3 and the • If v ∈ Ve (τ ) is of type definition of ǫ1 we have (R∞ Nc )2,D v 1 hv0 ≤ (R∞ Nc )2,D v 1 0 ≤ 2CNc 1 ≤ 2Cǫ1 = C2(hv −1)(2− |Pv | 2 −D v 1 ) ǫ1 , where the last equality holds trivially since we necessarily have hv = 2 and |Pv | = 2, and (R∞ Nc )2,D v 1 vanishes unless D v 1 = 2. • If v ∈ Ve (τ ) is of type (and, therefore, hv = 2), then int int (R∞ V∞ )|Pv |,D v 1 0 ≤ C |Pv | |λ|max{1,κ|Pv |} , )|Pv |,D v 1 hv0 ≤ (R∞ V∞ thanks to Lemma 4.3 and Eq.(3.15). • If v ∈ V0 (τ ), recalling the definition of W ∞ [υ; τv , P v , T v , Dv ], see Eq. (4.3.8), and the bounds on the norm of R∞ , see (4.2.22)–(4.2.25), we find (W ∞ [υ; τv , P v , T v , Dv ])|Pv |,D v 1 hv0 ′ ≤ C2−hv (D v 1 −D v 1 ) (W∞ [υ; τv , P v , T v , D′v ])|Pv |,D ′v 1 hv . (4.4.11)  Recalling the definition of D ′v , that is D ′v = w∈Sv D w P , as well as the v one of Rv , see the line before (4.3.9), we recognize that D v 1 −D ′v 1 = Rv . Note also that W∞ [υ; τv , P v , T v , D′v ] coincides with its restriction (W∞ [υ; τv , P v , T v , D′v ])|Pv |,D ′v 1 , so that the second line of (4.4.11) can be rewritten more compactly as C2−hv Rv W∞ [υ; τv , P v , T v , D′v ]hv . 1 1106 G. Antinucci et al. Plugging these bounds in (4.4.9), noting that we find W∞ [υ; τ, P , T , D] hv0 ≤ 1 |Sv0 |! C |Qv0 | +|Sv0 | 2 ⎧ |Pv | ⎪2(hv −1)(2− 2 −D v 1 ) ǫhv −1  ⎨ · C |Pv | |λ|max{1,κ|Pv |} ⎪ v∈Sv0 ⎩ −hv Rv W∞ [υ; τv , P v , T v , D′v ] 2 2 hv Ann. Henri Poincaré  z∈Λ∞ hv0 ( e− c0 2  |Qv0 | + v∈Sv 2 0 2hv0 z1 ≤ C2−2hv0 , D v |Qv 1 +2−2|Sv0 |) 0 · if v ∈ Ve (τ ) is of type if v ∈ Ve (τ )is of type if v ∈ V0 (τ ), (4.4.12) Now, in the last line, for v ∈ Sv0 ∩V0 (τ ), we iterate the bound, and we continue to do so until we reach all the endpoints. By doing so, recalling that Rv0 = 0, we obtain the desired bound, (4.4.2), provided that C  v∈V0 (τ ) (  |Qv | 2 +|Sv |) ≤ (C ′ ) v∈Ve (τ ) |Pv | . (4.4.13) In order to prove this, note that, for any dotted v ∈ V0 (τ ), |Sv | ≤ 1 + |Q2v | ≤ |Qv |, because |Qv | ≥ max{2, 2(|Sv |−1)}; moreover, ifv0 is not dotted, |Sv0 | = 1 and |Qv0 | = 0. Therefore, recalling also that |Qv | = w∈Sv |Pw |−|Pv |, we find C  v∈V0 (τ ) ( |Qv | 2 +|Sv |) which implies (4.4.13). 3 ≤ C 1+ 2  v∈V0 |Qv | 3 ≤ C 1+ 2  v∈Ve (τ ) |Pv | ,  Next, we rearrange (4.4.2) in a different form, more suitable for summing over GN trees and their labels. Lemma 4.7. Under the same assumptions as Proposition 4.6,  1 2hv0 d(Pv0 ,D v0 ) W∞ [υ; τ, P , T , D]hv0 ≤ C v∈Ve (τ ) |Pv | |Sv0 |! ×   v∈V ′ (τ ) 1 (hv −hv′ )d(Pv ,D v )   2 |Sv |! v∈Ve (τ ) ǫhv −1 |λ|max{1,κ|Pv |} (4.4.14) if v is of type if v is of type where d(Pv , D v ) = 2− |P2v | −D v 1 is the scaling dimension of v, see (4.3.10). Note that, as observed in Remark 4.4, the scaling dimensions appearing at exponent in the product over v ∈ V ′ (τ ) are all negative, with the exception of the case that v is an endpoint such that hv′ = hv − 1. Note, however, that in such a case 2(hv −hv′ )d(Pv ,D v ) ≤ 2, which is a constant that can be reabsorbed in C v∈Ve (τ ) |Pv | , up to a redefinition of the constant C.  Proof. First note that, for all v ∈ V0 (τ ), |Qv | = w∈Sv |Pw | − |Pv |, so that,   recalling that Rv = D v 1 − w∈Sv D w Pv 1 , we can rewrite the factor  1  ( 2 |Qv |+ w∈Sv D w |Qv 1 −Rv +2−2|Sv |)hv in (4.4.2) as v∈V0 (τ ) 2  1  1  v0 w∈Sv |Pw |− 2 |Pv |+ w∈Sv D w 1 −D v 1 −2(|Sv |−1)) v∈V0 (τ ) ( 2 2h ·   1 1 v∈V0 (τ ) (hv −hv0 )( 2 w∈Sv |Pw |− 2 |Pv |+ w∈Sv D w 1 −D v 1 −2(|Sv |−1)) ·2 . (4.4.15) Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1107 Now, the factor in the first line can be further rewritten by noting that:     |Pw | − |Pv | = (4.4.16) |Pv | − |Pv0 |, v∈V0 (τ ) w∈Sv   w∈Sv v∈V0 (τ )  v∈Ve (τ ) D v 1 − D v 1   = v∈Ve (τ ) D v 1 − D v0 1 , (4.4.17) (|Sv | − 1) = |Ve (τ )| − 1. (4.4.18) v∈V0 (τ ) (The first two identities are ‘obvious’, due to the telescopic structure of the summand; the latter identity can be easily proved by induction.) Therefore,  1 v∈V0 (τ ) ( 2 2hv0  w∈Sv hv0 (2− =2 |Pw |− |P2v | +  w∈Sv |Pv | 0 −D v0  1 ) 2 D w 1 −D v 1 −2(|Sv0 |−1))  −hv0 (2− |P2v | −D v 1 ) 2 v∈Ve (τ )  . (4.4.19) Similarly, the exponent of the factor in the second line of (4.4.15) can be rewritten as    1 1  |Pw | − |Pv | (hw − hw′ ) 2 2 v∈V0 (τ ) +   (hw − hw′ ) w∈V0 (τ ) w>v0 +  =  v∈V0 (τw ) 1 1  |Pw | − |Pv | 2 2 w∈Sv  D w 1 − D v 1 − 2(|Sv0 | − 1) w∈Sv   D w 1 − D v 1 − 2(|Sv0 | − 1) w∈Sv = w∈Sv w∈V0 (τ ) v0 <w≤v |Pw | − D w 1 − (hw − hw′ ) 2 − 2 w∈V0 (τ ) w>v0  v∈Ve (τw )  |Pv | 2− − D v 1 2  . (4.4.20) Using (4.4.19) and (4.4.20) and recalling that 2 − we rewrite (4.4.15) as hv0 d(Pv0 ,D v0 ) (4.4.15) = 2  v∈V0′ (τ ) (hv −hv′ )d(Pv ,D v ) 2 |Pv | 2  − D v 1 ≡ d(Pv , D v ), 1108 G. Antinucci et al.  −(hv0 + 2 v∈Ve (τ ) = 2hv0 d(Pv0 ,D v0 ) Ann. Henri Poincaré v0 <w<v w∈V0  (τ ) (hw −hw′ ))d(Pv ,D v ) 2(hv −hv′ )d(Pv ,D v ) v∈V0′ (τ )    2−hv′ d(Pv ,D v ) . v∈Ve (τ ) By using this rewriting in (4.4.2) and noting that ⎛ 1  2(hv −1)(2− 2 |Pv |−D v 1 ) ǫhv −1 ⎝ 2−hv′ d(Pv ,D v ) · |λ|max{1,κ|Pv |} v∈Ve (τ ) ⎛  ǫhv −1 if ≤⎝ 2(hv −hv′ )d(Pv ,D v ) · |Pv | max{1,κ|Pv |} 2 |λ| if v∈V (τ ) e hv d(Pv ,D v ) (since if v is an endpoint of type , then 2 obtain the desired estimate, (4.4.14).  (4.4.21) ⎞ if v is of type ⎠ if v is of type ⎞ v is of type ⎠ vis of type (4.4.22) −|Pv | ≥ 2 ), we readily  As announced above, the bound (4.4.14) is written in a form suitable for summing over the GN trees and their labels, as summarized in the following lemma. Lemma 4.8. Under the same assumptions as Proposition 4.6, for any ϑ ∈ (h) (0, 1), there exists Cϑ > 0 such that, letting T∞;(N,M ) denote the subset of (h) T∞ whose trees have N endpoints of type and M endpoints of type ,     W∞ [υ; τ, P , T , D]h+1 2−ϑh (h) τ ∈T∞;(N,M ) P ∈P(τ ) T ∈S(τ,P ) D∈D(τ,P ) |Pv0 |=n D v0 1 =p ≤ CϑN +M |λ|N ′ max 2−ϑh ǫh′ h<h′ ≤1 M (4.4.23) 2h·d(n,p) . Remark 4.9. (Short memory property). The fact that this estimate holds with the factors of 2−ϑh included indicates that the contribution of trees covering a large range of scales are exponentially suppressed, a behavior referred to in previous works (e.g., [18]) as the ‘short memory property’. As we shall see below, taking advantage of this there is a way to choose the free parameters Z, t∗1 , β such that |ǫh | ≤ Kϑ |λ|2ϑh , see Propositions 4.10 and 4.11 below. Under this condition, Lemma 4.8 implies that Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry     (h) τ ∈T∞;(N,M ) P ∈P(τ ) T ∈S(τ,P ) D∈D(τ,P ) |Pv0 |=n D v0 1 =p 1109 W∞ [υ; τ, P , T , D]h+1 ≤ CϑN +M |λ|N +M 2hd(n,p) 2ϑh , (4.4.24) for all ϑ < 1 and N + M > 0. The factor 2ϑh in the right side is called the ‘short-memory factor’. Summing over N + M ≥ 1, this immediately implies 2 2 2 2 ≤ Cϑ |λ|2hd(n,p) 2ϑh . (4.4.25) 2(W∞ )n,p 2 h+1 Proof. Thanks to (4.4.14), the left side of (4.4.23) can be bounded from above by 2h·d(n,p) |λ|N  ′ max 2−ϑh ǫh′ ′ h<h ≤1  D∈D(τ,P ) D  v0 1 =p 2ϑhv v∈Ve (τ ) M 2−ϑh 1 ⎛ |Sv0 |!    C (h) τ ∈T∞;(N,M ) P ∈P(τ ) |Pv0 |=n  ⎝ v∈V ′ (τ )   |Pv | v∈Vep (τ ) T ∈S(τ,P ) ⎞ 1 (hv −hv′ )d(Pv ,D v ) ⎠ 2 |Sv |! (4.4.26) |λ|[κ|Pv |−1]+ , v∈Ve (τ ) v of type v of type where, in the last factor, [·]+ indicates the positive part. Note that, if either v ∈ V0′ (τ ) or v is an endpoint such that hv′ < hv −1, then the scaling dimension of v can be bounded uniformly in D v , i.e., d(Pv , D v ) ≤ min{−1, 2 − |Pv |/2}; if v is an endpoint such that hv′ = hv − 1, then the factor2(hv −hv′ )d(Pv ,D v ) is |P | smaller than 2 (and, therefore, it can be reabsorbed in C v∈Vep (τ ) v up to a redefinition of the constant C). Moreover, recall that the number of elements of D(τ, P ) is bounded by 10|V (τ )| , see Remark 4.5. Finally, the number of elements of S(τ, P ) is bounded by   |P | |Sv |!, (4.4.27) |S(τ, P )| ≤ C v∈Vep (τ ) v v∈V0 (τ ) see, e.g., [15, Lemma A.5]. Therefore, putting these observations together, we see that (4.4.26) can be bounded from above by 2 h·d(n,p) × |λ|  v∈V ′ (τ ) N 2  max 2 h<h′ ≤1 −ϑh′ ǫh′ M 2 ϑ(hmax −h) τ   v∈Ve (τ ) v of type  ′ (C ) P ∈P(τ ) (h) τ ∈T∞;(N,M ) (hv −hv′ ) min{−1,2− |Pv | } 2  2 ϑhv  |λ|  v∈Vep (τ ) |Pv | [κ|Pv |−1]+ × . v∈Ve (τ ) v of type (4.4.28) 1110 G. Antinucci et al. Ann. Henri Poincaré We can cancel the factor of 2−ϑh with a product of factors leading to an endpoint, and simplify the remaining bounds, to get    −ϑh (hv −hv′ ) min{−1,2− |P2v | } 2 2ϑhv 2 v∈V ′ (τ ) ≤  (hv −hv′ )(ϑ−1) |P6v | 2 v∈V ′ (τ )  v∈Ve (τ ) v of type . This leaves a variety of exponential factors which make it possible to control the sum over P , giving      ′ (hv −hv′ )(ϑ−1) |P6v | v∈Vep (τ ) |Pv | (C ) 2 v∈V ′ (τ ) (h) τ ∈T∞;(N,M ) P ∈P(τ )  |λ|[κ|Pv |−1]+ ≤ (C ′′ )N +M , v∈Ve (τ ) v of type see, e.g., [15, App.A.6.1], from which (4.4.23) follows.  We conclude this subsection by noting that, in order for the right side of the bound (4.4.23) to be summable over N, M uniformly in h, we need that ǫh is bounded and small, uniformly in h. In view of Lemma 4.8, this condition is (h) sufficient for the whole sequence of kernels V∞ , h ≤ 1, to be well defined. In the next subsection, we study the iterative definition of the running coupling constants and prove that they in fact remain bounded and small, uniformly in h, provided that the counterterms υ1 are properly fixed. 4.5. Beta Function Equation and Choice of the Counterterms The definition of the running coupling constants, (4.2.30), combined with the GN tree expansion for the effective potentials implies that the running coupling constants υ = {(νh , ζh , ηh )}h≤0 satisfy the following equation, for all h ≤ 0:  2h νh Fν,∞ + ζh Fζ,∞ + ηh Fη,∞ = L∞ W∞ [υ; τ ]. (4.5.1) (h) τ ∈T∞ More explicitly, using the definitions of Fν,∞ , Fζ,∞ , Fη,∞ (see the lines after (4.2.4) and after (4.2.11)), for any z ∈ Λ∞ ,   νh = 2−h (2ω) L∞ W∞ [υ; τ ] (ω, 0, z), (−ω, 0, z) , (h) ζh = 4ω  τ ∈T∞ (h) τ ∈T∞ ηh = 4  (h) τ ∈T∞  L∞ W∞ [υ; τ ] (ω, 0, z), (ω, ê1 , z) ,  L∞ W∞ [υ; τ ] (ω, 0, z), (−ω, ê2 , z) . (4.5.2) Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1111 In view of the iterative definition of the kernels W∞ [υ; τ ], see Eqs. (4.3.4) and (4.3.5), the right sides of these three equations can be naturally thought of as functions of υ or, better yet, of the restriction of υ to the scales larger than h. (1) Therefore, these are recursive equations for the components of υ: given V∞ ∗ (and, in particular, given t1 , β, Z), one can in principle construct the whole sequence υ. More precisely, since the definition of the right sides of Eq. (4.5.2) requires the summations over τ to be well defined, in view of Lemma 4.8, these recursive equations allow one to construct the running coupling constants only for the scales h such that maxh′ >h ǫh′ is small enough. As we shall soon see, given a sufficiently small λ, the quantity maxh′ >h ǫh′ stays small, uniformly in h, only for a special choice of the free parameters t∗1 , β, Z; to understand the appropriate choice of these parameters, it is helpful first to isolate those trees with only counterterm vertices, whose contribution in (4.5.2) is especially simple: (h) 1. For h < 0, T∞;(0,1) consists of two trees, one with a single counterterm endpoint v on scale hv = 2 > hv0 + 1, for which L∞ W∞ [υ, τ ] = L∞ R∞ Nc = 0, and one with a single counterterm endpoint on scale hv = hv0 + 1, for which we have L∞ W∞ [υ, τ ] = 2h νh Fν,∞ + ζh Fζ,∞ + ηh Fη,∞ , (h) 2. For h = 0, T∞;(0,1) consists of only a single tree τ . This tree gives a contribution involving Nc which can be calculated quite explicitly from Eq. (3.23) and Eq. (2.1.13) taking advantage of the fact that the latter is written in terms of the Fourier transformed fields: L∞ W∞ [υ, τ ] = L∞ Nc  1 t1 t∗1 1 − t1  1 Fζ,∞ Fν,∞ + − t2 − = Z 1 + t1 Z (1 + t1 )2 (1 + t∗1 )2 t t∗  2 − 2 Fη,∞ + (4.5.3) 2Z 2 (recall t∗2 = (1 − t∗1 )/(1 + t∗1 )) which we can put in the same form as the other scales by letting t1 1 1 − t1  1 t∗1 t2 − , ζ1 := 2ν1 := − , Z 1 + t1 Z (1 + t1 )2 (1 + t∗1 )2 t∗ t2 − 2. (4.5.4) η1 := 2Z 2 (h) 3. The contributions from the trees in T∞,(0,M ) (i.e., those with no interaction endpoints and M counterterm endpoints) vanish for all h ≤ 0 and M ≥ 2, as can be seen as follows. In this case, each vertex must be assigned exactly 2 field labels, and for any endpoint v the components of ω v must be both imaginary (corresponding to Nm , i.e., to hv = 2, hv′ = 1) or both real (in all the other cases). If any endpoints of the first type appear there is no allowed assignment of field labels to the corresponding tree (since the ξ fields this case represents can neither be 1112 G. Antinucci et al. Ann. Henri Poincaré external fields nor be contracted with φ fields to form a connected diagram). In any case, all of the internal field labels are included in spanning trees, and the local part of the contribution can be made quite explicit.  (h) For example, denoting by g∞ (z − z ′ ) ω,ω′ the elements of the transla(h) tion invariant infinite volume propagator g∞ (z − z ′ ), see (2.2.4), (2.2.5), (2.2.9), the local part of the term with D v = 0 for all vertices (which contributes to νh ) is a sum of terms proportional to    (h1 ) 2) g∞ (z1 − z2 ) −ω ,ω g(h ∞ (z2 − z3 ) −ω ,ω 1 2 2 3 z2 ,...,zM ∈Z2  M −1 ) · · · g(h (zM −1 − zM ) −ω ∞ M −1 ,ω1    (h2 ) (h1 ) M −1 ) = ĝ∞ −ω ,ω (0) ĝ∞ −ω ,ω (0) · · · ĝ(h ∞ −ω 1 2 2 3 M −1 ,ω1 (0), (4.5.5) for some tuple (ω1 , . . . , ωM −1 ) ∈ {+, −}M −1 ; from Eqs. (2.2.9), (2.2.4) and (2.2.5), the Fourier transform ĝ(h) in the right side satisfies  2−2h (h) (4.5.6) ĝ (k) = ĝ[η] (k) dη, ĝ[η] (k) = e−ηD(k) D(k)ĝ(k) 2−2h−2 (except for h = 0, where the lower limit of integration is 0), with D defined in Eq. (2.1.21), from which it is evident that D(0) = 0, and in consequence that the right-hand side of Eq. (4.5.5) is likewise zero. The same argument holds, mutatis mutandis, when D v does not vanish for all vertices of the tree, with some discrete derivatives appearing in Eq. (4.5.5). In view of these properties, we can write ν νh = 2νh+1 + Bh+1 [υ], ζ ζh = ζh+1 + Bh+1 [υ], η ηh = ηh+1 + Bh+1 [υ], (4.5.7) ♯ for all h ≤ 0, where the functions Bh+1 [υ] are given by the restrictions of the (h) sums on the right-hand sides of Eq. (4.5.2) to trees in some T∞,(M,N ) with ♯ N ≥ 1. The functions Bh+1 [υ], with ♯ ∈ {ν, ζ, η}, are called the components of the beta function, and (4.5.7) are called the beta function flow equations for the running coupling constants; note that, even if not explicitly indicated, the ♯ functions Bh+1 [υ], in addition to υ, depend analytically upon λ, t∗1 , β, Z. For ♯ later reference, we let Bh♯ [υ; τ ] be the contribution to Bh+1 [υ] associated with the GN tree τ . In view of Lemma 4.8, we find that, for any |Z − 1| ≤ 1/2, any |t∗1 |, |t1 |, |t2 | ∈ K ′ , any ϑ ∈ (0, 1) there exists Cϑ > 0 such that, if |λ| and maxh′ >h {ǫh′ } are small enough, then max |Bh♯ [υ]| ≤ Cϑ |λ|2ϑh , ♯∈{ν,ζ,η} ∀ h ≤ 0. (4.5.8) Proposition 4.10. For any ϑ ∈ (0, 1), there exist Kϑ , λ0 (ϑ) > 0 and functions ν1 (λ; t∗1 , β, Z), ζ1 (λ; t∗1 , β, Z), η1 (λ; t∗1 , β, Z), υ(λ; t∗1 , β, Z) := {(νh (λ; t∗1 , β, Z), ζh (λ; t∗1 , β, Z), ηh (λ; t∗1 , β, Z))}h≤0 , analytic in |λ| ≤ λ0 (ϑ), |Z − 1| ≤ 1/2, Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1113 |t∗1 | ∈ K ′ and β such that |t1 |, |t2 | ∈ K ′ (recall that ti = tanh(βJi )), for all λ, t∗1 , β, Z in such an analyticity domain: 1. ν1 (λ; t∗1 , β, Z), ζ1 (λ; t∗1 , β, Z), η1 (λ; t∗1 , β, Z) and the components of υ(λ; t∗1 , β, Z) satisfy (4.5.7), for all h ≤ 0; 2. For all h ≤ 1, ǫh (λ; t∗1 , β, Z) := max{|νh (λ; t∗1 , β, Z)|, |ζh (λ; t∗1 , β, Z)|, |ηh (λ; t∗1 , β, Z)|} ≤ Kϑ |λ| 2ϑh . (4.5.9) Note that, at this point, we do not prove that ν1 (λ; t∗1 , β, Z), ζ1 (λ; t∗1 , β, Z), η1 (λ; t∗1 , β, Z) satisfy Eq. (4.5.4); this comes later. Proof. For simplicity, we do not track the dependence of the constants and of the norms upon ϑ which, within the course of this proof, we assume to be a fixed constant in (0, 1). In order to construct υ̃(λ; t∗1 , β, Z) := {(νh (λ; t∗1 , β, Z), ζh (λ; t∗1 , β, Z), ηh (λ; t∗1 , β, Z))}h≤1 , we first note that the equations for νh , ζh , ηh in (4.5.7) imply that, for k < h ≤ 1, νh = 2k−h νk −    j−h−1 ν Bj [υ], ζh = ζk − k<j≤h Bjζ [υ], and ηh = ηk − k<j≤h Bjη [υ]. If k<j≤h 2 we send k → −∞ and impose that ǫk := max{|νk |, |ζk |, |ηk |} → 0 as k → −∞, we get ⎧  j−h−1 ν ⎪ Bj [υ], ⎨νh = −  j≤h 2 ζ (4.5.10) ζh = − j≤h Bj [υ], ⎪  ⎩ ηh = − j≤h Bjη [υ], which we regard as a fixed point equation υ̃ = T [υ̃] for a map T on the space of sequences Xε := {υ̃ = {(νh , ζh , ηh )}h≤1 : υ̃ ≤ ε}, with υ̃ = suph≤1 {2−ϑh ǫh } and ε a sufficiently small constant. We now intend to prove that T is a contraction on Xε and, more precisely, that: (1) the image of Xε under the action of T is contained in Xε ; (2) T [υ̃] − T [υ̃ ′ ] ≤ (1/2) υ̃ − υ̃ ′  for all υ̃, υ̃ ′ ∈ Xε . Once T is proved to be a contraction, it follows that it admits a unique fixed point in Xε , which corresponds to the desired sequence υ̃(λ; t∗1 , β, Z). The analyticity of υ̃(λ; t∗1 , β, Z) follows from the analyticity of the components of the beta function with respect to λ, t∗1 , β, Z and υ that, in turn, follows from the absolute summability of its tree expansion, which is a power series in λ, with coefficients analytically depending upon t∗1 , t1 , t2 , Z, with ti = tanh(βJi ). The fact that the image of Xε under the action of T is contained in Xε follows immediately from Eq. (4.5.8). In order to prove that T [υ̃] − T [υ̃ ′ ] ≤ (1/2) υ̃ − υ̃ ′ , we rewrite the ν-component of T [υ̃]−T [υ̃ ′ ] at scale h as a linear interpolation  1   d ν νh − νh′ = − Bj [υ̃(t); τ ] dt, (4.5.11) 2j−h−1 0 dt (j−1) j≤h τ ∈T∞ ′ where υ̃(t) = υ̃ + t(υ̃ − υ̃ ′ ), and similarly for the ζ- and η-components. When the derivative with respect to t acts on the tree value Bjν [υ(t); τ ], it has the effect of replacing one of the factors νh (t), or ζh (t), or ηh (t), associated with 1114 G. Antinucci et al. Ann. Henri Poincaré d d one of the counterterm endpoints, by dt νh (t) = νh −νh′ , or dt ζh (t) = ζh −ζh′ , or d ′ dt ηh (t) = ηh − ηh , respectively. Therefore, we get the analogue of Eq. (4.5.8): if |λ| and υ̃(t) are sufficiently small, then    d ♯  max  Bh [υ(t); τ ] dt ♯∈{ν,ζ,η} (h−1) τ ∈T∞ ≤ C|λ|2ϑh max {|νh′ − νh′ ′ |, |ζh′ − ζh′ ′ |, |ηh′ − ηh′ ′ |} ′ h ≥h (4.5.12) for all h ≤ 1. Plugging this estimate into Eq. (4.5.11) and its analogues for ζh − ζh′ and ηh − ηh′ , we readily obtain the desired estimate, T [υ̃] − T [υ̃ ′ ] ≤  (1/2) υ̃ − υ̃ ′ , for λ0 and ε sufficiently small. We now need to show that it is possible to choose the free parameters β, Z, t∗1 in such a way that the functions ν1 (λ; t∗1 , β, Z), ζ1 (λ; t∗1 , β, Z), η1 (λ; t∗1 , β, Z) constructed in Proposition 4.10 satisfy Eq. (4.5.4); that is, for given J1 , J2 and λ, there exists a critical value of the inverse temperature β for which the above expansion for the kernels of the infinite plane effective potentials is convergent, with dressed parameters t∗1 and Z. The desired result is summarized in the following proposition. Proposition 4.11. For any J1 , J2 satisfying the conditions of Theorem 1.1, and any ϑ ∈ (0, 1), there exist λ0 (ϑ) > 0 and functions t∗1 (λ), βc (λ), Z(λ), analytic in |λ| ≤ λ0 (ϑ), such that Eq. (4.5.4) holds, with t1 = tanh(βc (λ)J1 ), t2 = tanh(βc (λ)J2 ), and (ν1 , ζ1 , η1 ) = ((ν̃1 (λ), ζ̃1 (λ), η̃1 (λ)), with ν̃1 (λ) = ν1 (λ; t∗1 (λ), βc (λ), Z(λ)) (here ν1 (λ; t∗1 , β, Z) is the same as in Proposition 4.10), and similarly for ζ̃1 (λ) and η̃1 (λ). Correspondingly, the flow of running coupling constants with initial datum (ν̃1 (λ), ζ̃1 (λ), η̃1 (λ)), generated by the flow equations (4.5.7), is well defined for all h ≤ 0 and satisfies (4.5.9). Proof. The result is a direct consequence of the analytic implicit function theorem: we intend to fix t∗1 = t∗1 (λ), β = βc (λ), Z = Z(λ) in such a way that Eq. (4.5.4) holds, with ν1 = ν1 (λ; t∗1 , β, Z), ζ1 = ζ1 (λ; t∗1 , β, Z), η1 = η1 (λ; t∗1 , β, Z), the same functions as in Proposition 4.10. With this in mind, we recast the system of equations (4.5.4) in the following form (recall once more that ti = tanh(βJi ) and t∗2 = (1 − t∗1 )/(1 + t∗1 )): 2Zν1 (λ; t∗1 , β, Z) − tanh βJ2 + e−2βJ1 = 0, 4Zt∗1 + e−4βJ1 − 1 = 0, 4Zζ1 (λ; t∗1 , β, Z) + (1 + t∗1 )2 1 − t∗1 2Zη1 (λ; t∗1 , β, Z) + Z − tanh βJ2 = 0. 1 + t∗1 (4.5.13) (4.5.14) (4.5.15) Note that, by Proposition 4.10, everything appearing in Eqs. (4.5.13) to (4.5.15) is analytic, so if we succeed in showing that the implicit function theorem applies, the resulting solution will be analytic in λ, as desired. Note that, at λ = 0, we have ν1 (0; t∗1 , β, Z) = ζ(0; t∗1 , β, Z) = η1 (0; t∗1 , β, Z) = 0, so in this case the system (4.5.13)–(4.5.15) is solved by β = βc (J1 , J2 ) (with Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1115 βc (J1 , J2 ) the critical temperature of the nearest neighbor model, see the  lines after (1.3)), t∗1 = tanh βc (J1 , J2 )J1 and Z = 1. Note also that the partial derivatives of ν1 (λ; t∗1 , β, Z), ζ1 (λ; t∗1 , β, Z), η1 (λ; t∗1 , β, Z) with respect to t∗1 , β and Z vanish at λ = 0, so the determinant of the Jacobian of the with respect to t∗1 , β, Z, computed at λ = 0 and system (4.5.13)–(4.5.15)  ∗ (t1 , β, Z) = (tanh βc (J1 , J2 )J1 , βc (J1 , J2 ), 1), equals    0 −J2 sech2 (βc J2 ) − 2J1 e−2βc J1 0    1−t∗1  4t∗ 1  4 (1+t∗ )3 −4J1 e−4βc J2 ∗ )2  (1+t   1 1   1−t∗ 1  − (1+t2 ∗ )2 −J2 sech2 (βc J2 ) ∗ 1+t 1 1 + (t∗1 )2 = −4(J2 sech2 (βc J2 ) + 2J1 e−2βc J1 ) (1 + t∗1 )4 1 (4.5.16) with βc = βc (J1 , J2 ) and t∗1 = tanh(βc (J1 , J2 )J1 ); the right-hand side is evidently nonzero, and, therefore, the analytic implicit function theorem applies, implying the desired claim.  This concludes the construction of the sequence of effective potentials in the infinite volume limit, uniformly in the scale label, with optimal bounds on the speed at which, after proper rescaling, such effective potentials go to zero as h → −∞. This result, and the methods introduced to prove it, is a key ingredient in the proof of Theorem 1.1, for whose completion we refer the reader to [4]. Acknowledgements We thank Hugo Duminil-Copin for several inspiring discussions. This work has been supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC CoG UniCoSM, grant agreement No. 724939 for all three authors and also ERC StG MaMBoQ, grant agreement No. 802901 for R.L.G.). G.A. acknowledges financial support from the Swiss Fonds National. A.G. acknowledges financial support from MIUR, PRIN 2017 project MaQuMA PRIN201719VMAST01. Funding Open access funding provided by Università degli Studi Roma Tre within the CRUI-CARE Agreement. Open Access. 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Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. A. Diagonalization of the Matrix Ac In this section we compute the propagator of φ and the Gaussian integral associated with Sc . For this purpose, we first need to block diagonalize the coefficient matrix, which we do by using a transformation which can be thought of as a Fourier sine transformation with modified frequencies. We write      1  −i∆(k1 ) −b(k1 ) + t2 τ φ+ (k1 ) φ+ (−k1 ) , Sc (φ) = φ− (k1 ) φ− (−k1 ) i∆(k1 ) b(k1 ) − t2 τ T 2L k1 ∈DL      1  φ+ (k1 ) φ+ (−k1 ) =: , (A.1) , Ãc (k1 ) φ− (k1 ) φ− (−k1 ) 2L k1 ∈DL where φω (k1 ), with ω = ±, is the column vector whose components are φω,z2 (k1 ) with z2 = 1, . . . , M , and τ is the M × M shift matrix τz2 ,z2′ := δz2 +1,z2′ , that is, ⎤ ⎡ 0 1 0 ⎥ ⎢ .. ⎥ ⎢0 . 0 1 ⎥ ⎢ ⎥ ⎢ . . . . .. .. .. .. ⎥ ⎢0 ⎥. ⎢ τ =⎢ ⎥ .. .. .. ⎢ . . . 1 0⎥ ⎥ ⎢ ⎥ ⎢ .. ⎣ . 0 0 1⎦ 0 0 0 For brevity, we will write Ãc = Ãc (k1 ), ∆ = ∆(k1 ), b = b(k1 ) since dependence on k1 plays no role in the next several pages. It is helpful to begin by diagonalizing the real symmetric matrix   + B̃c 0 2 Ãc = , 0 B̃c− where B̃c+ is the M × M tri-diagonal matrix + B̃c ⎡ bt2 −∆2 − b2 − t22 ⎢ bt2 −∆2 − b2 − t22 ⎢ ⎢ ⎢ ⎢ bt2 =⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ bt2 −∆2 − b2 − t22 .. . .. . .. . bt2 .. . −∆ − b2 − t22 bt2 2 bt2 −∆2 − b2 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎦ Note that all the diagonal entries are equal to −∆2 −b2 −t22 apart from the last one, which equals −∆2 − b2 . The block B̃c− is obtained from B̃c+ by reversing Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1117 the order of the rows and columns. B̃cω , with ω = ±, can each be thought of as a discrete Laplacian with mixed boundary conditions, which suggests the ansatz ⎤ ⎡ αk2 eik2 + βk2 e−ik2 i2k2 ⎢ + βk2 e−i2k2 ⎥ vk2 = ⎣αk2 e (A.2) ⎦, .. . for their eigenvectors. In fact, we see that vk2 is an eigenvector of B̃c+ iff the system of equations (−∆2 − b2 − t22 )(αk2 eik2 + βk2 e−ik2 ) + bt2 (αk2 ei2k2 + βk2 e−i2k2 ) = λk2 (αk2 eik2 + βk2 e−ik2 ) (A.3) bt2 (αk2 ei(z2 −1)k2 + βk2 e−ik2 (z2 −1) ) + (−∆2 − b2 − t22 )(αk2 eik2 z2 + βk2 e−ik2 z2 ) + bt2 (αk2 eik2 (z2 +1) + βk2 e−ik2 (z2 +1) ) = λk2 (αk2 eik2 z2 + βk2 e−ik2 z2 ), 1 < z2 < M (A.4) bt2 (αk2 eik2 (M −1) + βk2 e−ik2 (M −1) ) + (−∆2 − b2 )(αk2 eik2 M + βk2 e−ik2 M ) = λk2 (αk2 eik2 M + βk2 e−ik2 M ) (A.5) are all satisfied. Equation (A.4) is solved by choosing λk2 = bt2 (eik2 + e−ik2 ) + (−∆2 − b2 − t22 ), which reduces the other two conditions to bt2 (αk2 + βk2 ) = 0, ik2 (M +1) bt2 (αk2 e (A.6) −ik2 (M +1) + βk2 e )− t22 (αk2 eik2 M −ik2 M + βk2 e ) = 0. (A.7) The first condition implies βk2 = −αk2 , which can be used to rewrite Eq. (A.7) as sin k2 (M + 1) = B(k1 ) sin k2 M (A.8) where for brevity we have introduced B(k1 ) := |1 + t1 eik1 |2 t2 = t2 , b(k1 ) 1 − t21 (A.9) (cf. Eq. (2.1.22)). We call Q+ M (k1 ) the set of the solutions of (2.1.23) with ℜk2 ∈ [0, π]. Restricting to the critical case (2.1.17), B(k1 ) = 1 − 2t1 t2 (1 − cos k1 ) =: 1 − κ(1 − cos k1 ), 1 − t21 (A.10) so that 0<B(k1 )<1 for k1 ∈ DL . Equation (2.1.23) is equivalent to tan k2 (M + 1) = B(k1 ) sin k2 , B(k1 ) cos k2 − 1 (A.11) 1118 G. Antinucci et al. Ann. Henri Poincaré which for 0 <B(k1 ) < 1 has a unique real solution in each interval In := π 1 M +1 (n + 2 , n + 1), n = 0, . . . , M − 1, since the left-hand side increases monotonically from −∞ to 0, while the right-hand side is negative and decreasing. Thus, all M eigenvectors of B̃c+ (and, as a consequence, of B̃c− ) correspond to real solutions of this form by ⎤ ⎡ sin k2 : ⎢ sin 2k2 ⎥ 2 ⎥ ⎢ = u+ ⎥ ⎢ .. k2 NM (k1 , k2 ) ⎣ ⎦ . sin k2 M ⎤⎞ ⎛ ⎡ sin k2 M : ⎜ ⎢sin k2 (M − 1)⎥⎟ 2 ⎥⎟ ⎢ ⎜ ⎥⎟ , ⎜and, respectively, u− ⎢ .. k2 = NM (k1 , k2 ) ⎣ ⎦⎠ ⎝ . sin k2 (A.12) where NM (k1 , k2 ) := 2 M  sin2 k2 x = M + x=1 1 1 sin(2M + 1)k2 − 2 2 sin k2 (A.13) so that the eigenvectors are normalized. To obtain Eq. (2.1.24), we note that since q satisfies (A.8), we have sin [k2 (M + 1) ± k2 M ] = sin k2 (M + 1) cos k2 M ± sin k2 M cos k2 (M + 1) = [B(k1 ) cos k2 M ± cos k2 (M + 1)] sin k2 M, and so we can rewrite NM (k1 , k2 ) as NM (k1 , k2 ) = = B(k1 )M cos k2 M − (M + 1) cos k2 (M + 1) B(k1 ) cos k2 M − cos k2 (M + 1) d dk2 (B(k1 ) sin k2 M − sin k2 (M + 1)) B(k1 ) cos k2 M − cos k2 (M + 1) . (A.14) We now return to Ãc . Equation (2.1.23) is equivalent to and therefore beik2 (M +1) − t2 eik2 M = be−ik2 (M +1) − t2 e−ik2 M , (A.15) ik2 (M +1) − t2 eik2 M )u− (b − t2 τ T )u+ k2 , k2 = −(be and ik2 (M +1) − t2 eik2 M )u+ (−b + t2 τ )u− k2 k2 = (be whenever k2 ∈ QM (k1 ). Combining this with the definition of Ãc (k1 ) in Eq. (A.1), we see that  +   −i∆u+ uk2 k2 Ãc , = 0 −(beik2 (M +1) − t2 eik2 M )u− k2    ik (M +1)  0 (be 2 − t2 eik2 M )u+ k2 , and Ãc − = (A.16) u k2 i∆u− k2 Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1119  +   0 uk2 or in other words, the change of variables induced by and − puts u k2 0 Ãc in block-diagonal form, with 2 × 2 blocks g̃   −i∆ e−ik2 (M +1) (b − t2 eik2 ) −eik2 (M +1) (b − t2 e−ik2 ) i∆   1 −2it1 sin k1 e−ik2 (M +1) (1 − t21 )(1 − B(k1 )eik2 ) , = ik (M +1) 2 −ik 2 2 2 ik (1 − t1 )(1 − B(k1 )e ) 2it1 sin k1 |1 + t1 e 1 | −e −1 (k1 , k2 ) := (A.17) recalling the definitions (2.1.14) and (2.1.22). This block-diagonalization implies that   = PfAc = det g̃−1 (k1 , k2 ), (A.18) k1 ∈DL k2 ∈Q+ (k1 ) M where explicitly, using the criticality condition (2.1.17), det g̃−1 (k1 , k2 ) = 2(1 − t2 )2 (1 − cos k1 ) + 2(1 − t1 )2 (1 − cos k2 ) |1 + t1 eik1 | 2 . (A.19) Note that this determinant vanishes iff k1 = k2 = 0 mod 2π (in particular, it is positive if k1 ∈ DL ). Concerning the propagator, denoting the inverse of (A.17) by g̃(k1 , k2 ) := :=   g̃++ (k1 , k2 ) g̃+− (k1 , k2 ) g̃−+ (k1 , k2 ) g̃−− (k1 , k2 ) 1 D(k1 , k2 )   −e−ik2 (M +1) (1 − t21 )(1 − B(k1 )eik2 ) 2it1 sin k1 , eik2 (M +1) (1 − t21 )(1 − B(k1 )e−ik2 ) −2it1 sin k1 (A.20) where D(k1 , k2 ) is defined as in (2.1.21) and, letting φ̃k2 ,ω (k1 ) := M  φω,x2 (k1 )uω k2 (z2 ), z2 =1 we have, for k1 , k1′ ∈ DL , k2 , k2′ ′ ∈ Q+ M , and ω, ω ∈ {±}, (A.21) φ̃k2 ,ω (k1 )φ̃k2′ ,ω′ (k1′ ) = −Lδk1 ,−k1′ δk2 ,k2′ g̃ωω′ (k1 , k2 ),  so that, in terms of φω,z = L1 k1 ∈DL e−ik1 z1 φω,z2 (k1 ),  ′ 1  ω′ ′ φω,z φω′ ,z′  = − g̃ωω′ (k1 , k2 )e−ik1 (z1 −z1 ) uω k2 (z2 )uk2 (z2 ) L + k1 ∈DL k2 ∈Q (k1 ) M ≡ gωω′ (z, z ′ ). u± k2 (z2 ) (A.22) Then recalling the definition of and the identity (A.15), Eqs. (2.1.18) and (2.1.19) follow by writing out the sines in terms of complex exponentials + and relabeling the sum in terms of QM (k1 ) := Q+ M (k1 ) ⊔ (−QM (k1 )). 1120 G. Antinucci et al. Ann. Henri Poincaré B. Proof of Proposition 2.3 For the proof of items 1 and 2 it is convenient to start by proving their analogues for the infinite volume limit propagators. [η] (h) [η] Decay bounds on g∞ , g∞ . Recall that g∞ was defined in (2.2.9). We intend to prove that, for all x ∈ Z2 , [η] (z) ≤ C 1+r+s × ∂1r ∂2s g∞ r!s!η − e−|z|1 3+r+s 2 e−η −1/2 |z|1 η ≥ 1, 0 ≤ η ≤ 1, if if (B.1) where ∂j is the discrete derivative with respect to the j-th coordinate. By using (h) (B.1) in the definition of g∞ , namely ⎧ 1 ⎪ [η] ⎪ g∞ (z) dη, if h = 0, ⎨ (h) 0 g∞ (z) =  2−2h (B.2) ⎪ ⎪ [η] ⎩ g∞ (z) dη, if h < 0, 2−2h−2 we obtain the analogue of (2.2.15), (h) ∂1r ∂2s g∞ (z) ≤ C 1+r+s r!s!2(1+r+s)h e−c2 h |z|1 , (B.3) for all z ∈ Z2 and h ≤ 0. In order to prove (B.1), we start from the explicit expression of the function in the left side, ∂1r ∂2s g[η] ∞ (z)  dk1 dk2 e−i(k1 z1 +k2 z2 ) (e−ik1 − 1)r (e−ik2 − 1)s e−ηD(k1 ,k2 ) M (k1 , k2 ) , = (2π)2 [−π,π]2 (B.4) where D(k1 , k2 ) is as in Eq. (2.1.21) (cf. Eq. (2.2.1)) and M (k1 , k2 ) := D(k1 , k2 )ĝ(k1 , k2 )   −2it1 sin k1 −(1 − t21 )[1 − e−ik2 B(k1 )] = . (1 − t21 )[1 − eik2 B(k1 )] 2it1 sin k1 (B.5) Note that the integrand in Eq. (B.4) is periodic with period 2π both in k1 and in k2 , and is entire in both arguments. Therefore, we can shift both variables in the complex plane, k1 → k1 − ia and k2 → k2 − ib, to obtain  [η] ∂1r ∂2s g∞ (z) = e−az1 −bz2 e−i(k1 x1 +k2 x2 ) (e−ik1 −a − 1)r (e−ik2 −b − 1)s [−π,π]2 −ηD(k1 −ia,k2 −ib) ×e (B.6) dk1 dk2 . M (k1 − ia, k2 − ib) (2π)2 Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1121 We now pick a = α sign(z1 ) and b = α sign(z2 ), with α = min{η −1/2 , 1}, and take the absolute value, thus getting 2 2 [η] (z) ≤ e−α|z|1 e−2η[(1−t1 ) +(1−t2 ) ](cosh α−1) ∂1r ∂2s g∞  × e−ηD(k1 ,k2 ) (αeα + |e−ik1 − 1|)r (B.7) [−π,π]2 (αeα + |e−ik2 − 1|)s M0 (k1 , k2 , α) dk1 dk2 , (2π)2 where M0 (k1 , k2 , α) = max|a|=|b|=α M (k1 − ia, k2 − ib) and we used the fact that, if |a| = |b| = α, then |D(k1 − ia, k2 − ib)| ≥ D(k1 , k2 ) + 2[(1 − t1 )2 + (1 − t2 )2 ](cosh α − 1) (B.8) and |e−ik1 −a − 1| ≤ αeα + |e−ik1 − 1|, |e−ik2 −b − 1| ≤ αeα + |e−ik2 − 1|. (B.9) Now, if η ≤ 1 and, therefore, α = 1, then (B.7) immediately implies [η] that ∂1r ∂2s g∞ (z) ≤ C 1+r+s e−|z|1 , as desired. If η ≥ 1 and, therefore, α = η −1/2 , we make the following observations: (i) the factor 2 2 −1/2 −1) e−2η[(1−t1 ) +(1−t2 ) ](cosh η is bounded from above uniformly in η; (ii) if −π ≤ k1 , k2 ≤ π, then D(k1 , k2 ) ≥ c(k12 + k22 ), |e−ik1 − 1| ≤ C|k1 |, |e−ik2 − 1| ≤ C|k2 | and M0 (k1 , k2 , η −1/2 ) ≤ C(η −1/2 + |k1 | + |k2 |). By using these inequalities in (B.7), we find −1/2 [η] |z|1 (z) ≤ C 1+r+s e−η ∂1r ∂2s g∞  2 2 × e−cη(k1 +k2 ) (η −1/2 + |k1 |)r (η −1/2 + |k2 |)s (η R2 −1/2 + |k1 | + |k2 |) dk1 dk2 , (B.10) and expanding the powers in the integrand we obtain a sum of Gaussian integrals which reduce to (B.1) for this case as well. [η] (h) [η] Decay bounds on gE , gE , and proof of items 1 and 2. Recall that gE (z, z ′ ) = [η] [η] g[η] (z, z ′ ) − gB (z, z ′ ), with g[η] as in (2.2.7) and gB as in (2.2.10). We focus ′ on the case that z1 − z1 = ±L/2 (recall that in our conventions z1 , z1′ ∈ {1, . . . , L}), in which the function sL in (2.2.10) is equal to ±1; the comple[η] mentary case, z1 − z1′ = ±L/2, in which gB (z, z ′ ) = 0, can be treated in a way analogous to the discussion below, and is left to the reader. If z1 − z1′ = ±L/2, by using the anti-periodicity of the propagator in the horizontal direction, we can reduce without loss of generality to the case z1 − z1′ = perL (z1 − z1′ ) (i.e., [η] [η] −L/2 < z1 − z1′ < L/2), in which gB (z, z ′ ) = g∞ (z − z ′ ), and we shall do so in the following. Therefore, in this case, [η] gE (z, z ′ ) =   k1 ∈DL k2 ∈QM (k1 )   1 [η] [η] G+ (k1 , k2 ; z, z ′ ) − G− (k1 , k2 ; z, z ′ ) 2LNM (k1 , k2 ) 1122 G. Antinucci et al. Ann. Henri Poincaré ′ −g[η] ∞ (z − z ), (B.11) where, recalling the definitions of fη in (2.2.1) and of ĝ(k1 , k2 ) and ĝωω′ (k1 , k2 ) in (2.1.20), [η] ′ ′ [η] ′ ′ G+ (k1 , k2 ; z, z ′ ) := e−ik1 (z1 −z1 ) e−ik2 (z2 −z2 ) fη (k1 , k2 )ĝ(k1 , k2 ), G− (k1 , k2 ; z, z ′ ) := e−ik1 (z1 −z1 ) e−ik2 (z2 +z2 ) fη (k1 , k2 )   ĝ++ (k1 , k2 ) ĝ+− (k1 , −k2 ) , ĝ−+ (k1 , k2 ) e2ik2 (M +1) ĝ−− (k1 , k2 ) (B.12) which are entire functions of k1 , k2 , and 2π-periodic in both variables. We intend to prove that, for all z, z ′ ∈ Λ, 3+|r |1 [η] ∂ r gE (z, z ′ ) ≤ C 1+|r |1 × r!η − 2 e−cη ′ e−c dE (z,z ) −1/2 dE (z,z ′ ) ∗ 1 ≤ η ≤ 2−2h , 0 ≤ η ≤ 1, if if (B.13) where we recall that h∗ = −⌊log2 min{L, M }⌋. Recalling also the relationship (h) [η] between gE and gE , this implies (h) ∂ r gE (z, z ′ ) ≤ C 1+|r|1 r!e−c2 h dE (z,z ′ )  2−2h η −(3+|r |1 )/2 dη 2−2h−2 (1+|r |1 )h −c2h dE (z,z ′ ) ≤ (C ′ )1+|r |1 r!2 e , (B.14) ′ (0) for h∗ < h < 0, and ∂ r gE (z, z ′ ) ≤ C 1+|r|1 e−c dE (z,z ) for h = 0. (h) (h) Recalling that g(h) = g∞ + gE and noting that dE (z, z ′ ) ≥ z − z ′ 1 , inequalities (B.3) and (B.14) also imply that g(h) satisfies a bound of the form (2.2.15). In order to prove (B.13), we start from (B.11). Recalling that QM (k1 ) is the set of roots of B(k1 ) sin k2 M − sin k2 (M + 1), with k2 ∈ (−π, π], that [η] NM (k1 , k2 ) is given by (A.14), and that G♯ (k1 , k2 ; z, z ′ ) are entire and 2πperiodic, for ♯ ∈ {±}, we can rewrite  k2 ∈QM (k1 ) 1 = 2 > C 1 [η] G (k1 , k2 ; z, z ′ ) 2NM (k1 , k2 ) ♯ B(k1 ) cos k2 M − cos k2 (M + 1) [η] dk2 G (k1 , k2 ; z, z ′ ) , B(k1 ) sin k2 M − sin k2 (M + 1) ♯ 2πi (B.15) where C is the boundary of the rectangle in the complex plane of vertices −π − ib, π − ib, π + ib, −π + ib, b > 0, traversed counterclockwise. We rewrite   R± (k1 , k2 )e±2ik2 (M +1) B(k1 ) cos k2 M − cos k2 (M + 1) = ∓i 1 + 2 , B(k1 ) sin k2 M − sin k2 (M + 1) 1 − R± (k1 , k2 )e±2ik2 (M +1) (B.16) Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1123 where 1 − B(k1 )e∓ik2 , 1 − B(k1 )e±ik2 (B.17) Rσ (k1 , k2 )e2iσk2 (M +1) . 1 − Rσ (k1 , k2 )e2iσk2 (M +1) (B.19) R± (k1 , k2 ) := and using this and noting that the contributions of the left and right sides of C in the integral in Eq. (B.15) cancel by the periodicity of the integrand, we then have  1 [η] G (k1 , k2 ; z, z ′ ) 2NM (k1 , k2 ) ♯ k2 ∈QM (k1 )   π+iσb dk2 [η] = (B.18) Aσ (k1 , k2 )G♯ (k1 , k2 ; z, z ′ ) 2π σ=0,±1 −π+iσb where A0 (k1 , k2 ) ≡ 1 and, if σ = ±1, Aσ (k1 , k2 ) := We now need to sum (B.18) over k1 ∈ DL . Notice that the right side of (B.18) is analytic in k1 in a sufficiently small strip around the real axis (quantitative bounds on the width of the analyticity strip will follow) and is 2π-periodic in k1 . Given any function F (k1 ) that is 2π-periodic and analytic in a strip of width 2b > 0 around the real axis, we have >   π+iσ′ b dk1 F (k1 ) dk1 1  A′σ′ (k1 ) F (k1 ) = , F (k1 ) = −ik1 L L 2π 1 + e 2π ′ −π+iσ b C ′ k1 ∈DL σ =0,±1 (B.20) A′0 (k1 ) ′ iσ ′ k1 L A′σ′ (k1 ) iσ ′ k1 L where ≡ 1 and, if σ = ±1, = −e /(1 + e ); moreover C is the same contour defined after (B.15). Using (B.11), (B.18) and (B.20), we obtain  dk1 dk2  r [η] [η] ∂ r gE (z, z ′ ) = − ∂ G− (k1 , k2 ; z, z ′ ) (2π)2 [−π,π]2 +  ♯=± ♯ ∗  [η] A′σ′ (k1 + iσ ′ b) Aσ (k1 + iσ ′ b, k2 + iσb) ∂ r G♯ σ,σ ′ =0,±  (k1 + iσ ′ b, k2 + iσb; z, z ′ ) , (B.21) ′ where the ∗ on the sum indicates the constraint that (σ, σ ) = (0, 0), and [η] ∂ r G♯ (k1 , k2 ; z, z ′ ) = (e−ik1 − 1)r1,1 (eik1 − 1)r2,1 (e−ik2 − 1)r1,2 [η] (e♯ik2 − 1)r2,2 G♯ (k1 , k2 ; z, z ′ ). [η] (B.22) Now, by using the definition of G− and by proceeding as in the proof of (B.1), we see that the first term in the right side of (B.21) admits the same bound as [η] g∞ (z − z ′ ), see (B.1), with the only difference that |z − z ′ |1 should be replaced 1124 G. Antinucci et al. Ann. Henri Poincaré by |z1 − z1′ | + min{z2 + z2′ , 2(M + 1) − z2 − z2′ } ≤ dE (z, z ′ ). Therefore, the first term in the right side of (B.21) satisfies (B.13) as desired. [η] Let us now prove that the contribution to [∂ r gE (z, z ′ )]ωω′ from the second line of (B.21) satisfy (B.13). For this purpose, if σ ′ = 0, we shift k1 in the complex plane as k1 → k1 −i b sign(z1 −z1′ ). If σ = 0, we shift k2 in the complex plane as k2 → k2 − iτ b, with τ = ±, its specific valued depending on ♯ and on the matrix element (ω, ω ′ ) we are looking at; more precisely, τ = τ♯,(ωω′ ) , with ⎧ ′ ⎪ ⎨−sign(z2 − z2 ) if ♯ = + τ♯,(ωω′ ) := −1 (B.23) if ♯ = − and (ω, ω ′ ) = (−, −) ⎪ ⎩ ′ +1 if ♯ = − and (ω, ω ) = (−, −). Once these complex shifts are performed, we bound the contribution to [η] [∂ r gE (z, z ′ )]ωω′ from the second line of (B.21) by the sum over ♯ and over σ, σ ′ (with (σ, σ ′ ) = (0, 0)) of:    dk1 dk2  ′ Aσ′ (k1 + iσ̃ ′ b) Aσ (k1 + iσ̃ ′ b, k2 + iσ̃b) 2 (2π) [−π,π]2    r [η]  [∂ G♯ (k1 + iσ̃ ′ b, k2 + iσ̃b; z, z ′ )]ωω′ , σ τ♯,(ωω′ ) that, if σ ′ =  0, then where σ̃ = if σ = 0 , and σ̃ ′ = if σ = 0 (B.24) σ′ −sign(z1 − z1′ ) if σ ′ = 0 . Note if σ ′ = 0 e−bL . (B.25) 1 − e−bL If σ = 0, we recall that Aσ (k1 , k2 ) is given by (B.19), with Rσ (k1 , k2 ) as in (B.17). We claim that, if b ≤ c0 , with c0 sufficiently small, and k1 , k2 real, then   Rσ (k1 + iσ̃ ′ b, k2 + iσ̃b) ≤ eCb . (B.26) |A′σ′ (k1 + iσ̃ ′ b)| ≤ for some C > 0; this will be proved momentarily, after (B.29). We now pick b = c0 min{1, η −1/2 }, so that, using (B.26), for σ =  0, M sufficiently large, and k1 , k2 real, eCb e−2b(M +1) . (B.27) 1 − e−bM If we now use (B.25) and (B.27) in the second line of (B.21) and we estimate   [η] the integral of ∂ r G♯ (k1 + iσ̃ ′ b, k2 + iσ̃b; z, z ′ ) via the same strategy used in the proof of (B.1), see Eqs. (B.7) to (B.10), we find that, for η ≤ 1, |Aσ (k1 + iσ̃ ′ b, k2 + iσ̃b)| ≤ ′ (B.24) ≤ C 1+|r |1 e−c0 dE (z,z ) , (B.28) ∗ where dE was defined after (2.2.16), while, if 1 ≤ η ≤ 2−2h , (B.24) ≤ C 1+|r |1 e−c0 η −1/2 ≤ (C ′ )1+|r |1 e−c0 η dE (z,z ′ ) −1/2 2 2 intR2 dk1 dk2 e−cη(k1 +k2 ) (|k1 | + |k2 | + η −1/2 )1+|r |1 dE (z,z ′ ) r !η − 3+|r |1 2 . (B.29) Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1125 This completes the proof that the terms in the second line of (B.21) satisfy (B.13), with c = c0 , provided that the bound (B.26) holds. Proof of (B.26). We will prove the following version of (B.26): if σ = 0, b ≤ c0 with c0 sufficiently small and a := σ̃ ′ b with σ̃ ′ = ±1, then |Rσ (k1 +ia, k2 + iσb)|2 ≤ e2b (1 + C0 b), (B.30) where C0 can be chosen 0 / C0 = max 8κ(1 + π 2 ), 128π 4 κ(1 − 2κ)−2 , (B.31) and κ is the same as in (A.10). By using the definition of Rσ , (B.17), one sees that (B.30) is equivalent to  e−2b + ρ2 − 2ρe−b cos(σk2 − β) ≤ (1 + C0 b) 1 + ρ2 e−2b − 2ρe−b cos(σk2 + β) , (B.32) where ρ = ρ(k1 , a) := |B(k1 +ia)| and β = β(k1 , a) := Arg(B(k1 +ia)). As shown below, if −π ≤ k1 ≤ π and |a|= b ≤ c0 with c0 sufficiently small, then ρ ≤ 1 + κ[(1 + π 2 )b2 − k12 /π 2 ] , |β| ≤ 4κ |k1 | · b . 1 − 2κ (B.33) By rearranging the terms in the two sides, one sees that (B.32) is equivalent to (ρ2 − 1)(1 − e−2b ) ≤ 4ρe−b sin(σk2 ) sin β + C0 b[(1 − ρe−b )2  +2ρe−b 1 − cos(σk2 + β) ]. (B.34) By using the first bound in (B.33), the fact that |a| = b ≤ c0 with c0 sufficiently small, and the bound 1 − cos(σk2 + β) ≥ (σk2 + β)2 /π 2 valid for β sufficiently small, it is straightforward to check that the left side of (B.34) is smaller or equal than 4κb[(1+π 2 )b2 −k12 /π 2 ], while the right side is greater or equal than   b2 2 + 2 ρe−b (|k2 | − |β|)2 . −4ρe−b |k2 | · |β| + C0 b 2 π Therefore, (B.34) is a consequence of 4κ(1 + π 2 )b3 + 4ρe−b |k2 | · |β| ≤ 4κbk12 /π 2 + C0 b3 2 + C0 b 2 ρe−b (|k2 | − |β|)2 . 2 π (B.35) Now, the first term in the left side of (B.35) is smaller than the third term in the right side, C0 b3 /2, because C0 ≥ 8κ(1 + π 2 ), see (B.31). By using the second bound in (B.33) and the fact that ρe−b ≤ 1 for b small enough (thanks to the first bound in (B.33)), we see that, if |k2 | ≤ 1−2κ 4π 2 |k1 |, then the second term in the left side of (B.35) is smaller than the first term in the right side. In the complementary case, |k2 | ≥ 1−2κ 4π 2 |k1 | (which implies, in particular, that |k2 | ≥ 2|β|, thanks to the second bound in (B.33)), then the second term in 1126 G. Antinucci et al. Ann. Henri Poincaré 16κ the left side of (B.35) is bounded from above by 1−2κ ρe−b b|k2 | · |k1 |, while the −b 2 0b last term in the right side is bounded from below by C 2π 2 ρe |k2 | ; now, 16κ C0 b −b ρe−b b|k2 | · |k1 | ≤ ρe |k2 |2 1 − 2κ 2π 2 ⇔ |k2 | ≥ 32π 2 κ |k1 |, C0 (1 − 2κ) 4 −2 which is verified for |k2 | ≥ 1−2κ , see 4π 2 |k1 |, because C0 ≥ 128π κ(1 − 2κ) (B.31). In conclusion, (B.35) is always verified and, as a consequence, (B.34) (and, therefore, (B.30)) is, as desired. We are left with proving the validity of (B.33) for |a|= b small enough. By definition ρeiβ = 1 − κ + κ cos k1 cosh a−iκ sin k1 sinh a, (B.36) so that, using 1 − cos k1 ≥ 2k12 /π 2 and the fact that, for |a|= b small, cosh a ≤ 1 + b2 and | sinh a| ≤ 2b, ρ ≤ 1 + κ(b2 − 2k12 /π 2 + 2b|k1 |). Using 2b|k1 | ≤ π 2 b2 + k12 /π 2 , we get the first of (B.33). Finally, from (B.36), we find κ| sin k1 sinh a| 2κb|k1 | |β| ≤ ≤ . 1 − κ + κ cos k1 cosh a 1 − κ(2 + b2 ) Now, picking b2 smaller than (1 − 2κ)/2κ, we find that β satisfies the second  of (B.33). Gram representation: proof of items 3 and 4. Recall that ′ (h) ∂ (s,s ) gωω′ (z, z ′ )    = dη  ′ 1 [η] ∂ (s,s ) G+,ωω′ (k1 , k2 ; z, z ′ ) 2LNM (k1 , k2 ) Ih k1 ∈DL k2 ∈QM (k)  (s,s ′ ) [η] G−,ωω′ (k1 , k2 ; z, z ′ ) , (B.37) −∂ where I0 = [0, 1), and Ih = [2−2h−2 , 2−2h ) for all h∗ ≤ h < 0. We recall ′ [η] that ∂ (s,s ) G♯ is given by (B.22), with s playing the role of (r1,1 , r1,2 ) and s′ playing the role of (r2,1 , r2,2 ). In the following, we will exhibit a Gram [η] decomposition separately for the two terms in (B.37) corresponding to G+ and [η] G− , which will immediately imply a Gram decomposition for the combination of the two. We rewrite ′ [η] 1Ih (η) ∂ (s ,s ) G♯,ωω′ (k1 , k2 ; z, z ′ ) = 4  σ=1 (h) γ̃♯,ω,s ,z (k1 , k2 , η) ∗ σ (h) γ♯,ω′ ,s ′ ,z′ (k1 , k2 , η)  σ , (B.38) where 1Ih is the characteristic function of the interval Ih , and   (h) (h) γ♯,ω,s,z (k1 , k2 , η) σ , γ̃♯,ω,s,z (k1 , k2 , η) σ are the components of the following Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 4-vectors: (h) γ̃♯,+,s,z (k1 , k2 , η) = 1Ih (η) eik1 z1 +ik2 z2 (eik1 − 1)s1 (eik2 − 1)s2 ⎡= ∗ ⎤ ĝ♯,++ (k1 , k2 ) ⎢= ∗ ⎥ ⎢ ⎥ ⎢ ĝ♯,+− (k1 , k2 ) ⎥ , ⎢ ⎥ ⎣ ⎦ 0 0 (h) γ̃♯,−,s,z (k1 , k2 , η) = 1Ih (η) eik1 z1 +ik2 z2 (eik1 − 1)s1 (eik2 − 1)s2 ⎤ ⎡ 0 ⎥ ⎢ 0 ⎢= ∗ ⎥ ⎥ ⎢ ⎢ ĝ♯,−+ (k1 , k2 ) ⎥ , ⎣= ∗ ⎦ ĝ♯,−− (k1 , k2 ) 3 fη (k1 , k2 ) 3 fη (k1 , k2 ) (h) γ♯,+,s,z (k1 , k2 , η) = 1Ih (η) eik1 z1 +♯ik2 z2 (eik1 − 1)s1 (e♯ik2 − 1)s2 ⎡= ⎤ ĝ♯,++ (k1 , k2 ) ⎢ ⎥ 0 ⎢ = ⎥ ⎣ ĝ♯,−+ (k, q) ⎦ , 0 (h) 1127 γ♯,−,s,z (k1 , k2 , η) = 1Ih (η) eik1 z1 +♯ik2 z2 (eik1 − 1)s1 (e♯ik2 − 1)s2 ⎡ ⎤ 0 = ⎢ ĝ♯,+− (k1 , k2 )⎥ ⎥, ⎢ ⎣ ⎦ 0 = ĝ♯,−− (k1 , k2 ) 3 fη (k1 , k2 ) 3 fη (k1 , k2 ) where if ♯ = +, then ĝ+,ωω′ (k1 , k2 ), with ω, ω ′ ∈ {±}, are the components of the 2×2 matrix ĝ+ (k1 , k2 ) ≡ ĝ(k1 , k2 ), see (2.1.20); if ♯ = −, then ĝ♯,ωω′ (k1 , k2 ), with ω, ω ′ ∈ {±}, are the components of   ĝ++ (k1 , k2 ) ĝ+− (k1 , −k2 ) , ĝ− (k1 , k2 ) ≡ ĝ−+ (k1 , k2 ) e2ik2 (M +1) ĝ−− (k1 , k2 ) = cf. (C.6). The square roots g♯,ωω′ (k1 , k2 ) of the complex numbers g♯,ωω′ (k1 , k2 ) are all defined by the same (arbitrarily chosen) branch. In conclusion, in light of (B.38), (B.37) can be rewritten as ′ (h) ∂ (s,s ) gωω′ (z, z ′ )  ∞  dη = 0  k1 ∈DL k2 ∈QM (k1 )  (h) γ♯,ω′ ,s ′ ,z′ (k1 , k2 , η) σ 4   ∗ 1 (h) ♯ γ̃♯,ω,s,z (k1 , k2 , η) σ 2LNM (k1 , k2 ) σ=1 ♯=±   (h) (h) (h) (h) =: γ̃+,ω,s,z ⊗ ê1 + γ̃−,ω,s,z ⊗ ê2 , γ+,ω′ ,s ′ ,z′ ⊗ ê1 − γ−,ω′ ,s ′ ,z′ ⊗ ê2 1128 G. Antinucci et al. Ann. Henri Poincaré   (h) (h) ≡ γ̃ω,s,z , γω′ ,s ′ ,z′ , (B.39) where in the last line ê1 , ê2 are the elements of the standard Euclidean basis of R2 . We can adapt all of the preceding discussion to g(≤h) simply by replacing Ih with [2−2h−2 , ∞); this concludes the proof of item 3. In order to prove the bounds in item 4, we first note that the definitions (h) (h) given above for γ̃ω,s,z , γω,s,z immediately imply        1  (h) 2  (h) 2 dη γ̃ω,s,z  , γω,s,z  ≤ LN (k M 1 , k2 ) Ih k∈DL k2 ∈QM (k) (B.40)     ik  e 1 − 12s1 eik2 − 12s2 · ·fη (k1 , k2 ) |gωω′ (k1 , k2 )|. ω,ω ′ =± Now, recall that the set DL consists of points in [−π, π] that are equi-spaced at a mutual distance 2π/L, and that the set QM (k) consists of points in [−π, π] that are almost equi-spaced at a mutual distance π/(M + 1) (more precisely, recall that there is exactly one point of QM (k) in every interval 1 π M +1 (n + 2 , n + 1), n = 0, . . . , M − 1, and exactly one point in every interval 1 π 11 M +1 (−n − 1, −n − 2 ), n = 0, . . . , M − 1). Note also that NM (k1 , k2 ) ≥ M , and that the summand in (B.40) is continuous, so we can bound (B.40) by a Riemann sum and obtain         (h) 2  (h) 2 dk1 dk2 eik1 dη γ̃ω,s,z  , γω,s,z  ≤ C Ih [−π,π]2  2s2 2s1  |gωω′ (k1 , k2 )| −1 eik2 − 1 fη (k1 , k2 ) ≤ (C ′ )1+s1 +s2  Ih dη ω,ω ′ =±  dk1 dk2 [−π,π]2 2 2 min(1, |k1 |2s1 |k2 |2s2 e−cη(k1 +k2 ) (|k1 | + |k2 |))  ≤ (C ′′ )1+s1 +s2 s1 !s2 ! min(1, η −s1 −s2 −3/2 ) dη Ih ≤ (C ′′′ )1+2s1 +2s2 s1 !s2 ! 2h(1+2s1 +2s2 ) . (B.41) Similarly       (≤h) 2  (≤h) 2 ′′ 1+s1 +s2 s1 !s2 ! γ̃ω,s,z  , γω,s,z  ≤(C ) ∞ η −s1 −s2 −3/2 dη 2−2h−2 ′′′ 1+2s1 +2s2 s1 !s2 ! 2h(1+2s1 +2s2 ) , ≤ (C ) (B.42) and these bounds constitute item 4. 11 In fact, by Eq. (A.13) and the definition of QM (k1 ), one has NM (k1 , k2 ) − M = (1 − B(k1 ) cos k2 )/(B 2 (k1 ) − 2B(k1 ) cos k2 + 1) ≥ 0. Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1129 C. Proof of Proposition 2.9 Recall that Λ is the discrete cylinder of sides L = 2⌊a−1 ℓ1 /2⌋ and M + 1 = ⌊a−1 ℓ2 ⌋+1 and gscal (z, z ′ ) the scaling limit propagator (2.3.3) in the continuum cylinder Λℓ1 ,ℓ2 of sides ℓ1 , ℓ2 . In order to emphasize its dependence upon the sides of the cylinder, let us denote the scaling limit propagator in Λℓ1 ,ℓ2 by gscal (ℓ1 , ℓ2 ; z, z ′ ). Note that, upon rescaling by ξ > 0, this propagator satisfies: ξgscal (ℓ1 , ℓ2 ; ξ z, ξ z ′ ) = gscal (ξ −1 ℓ1 , ξ −1 ℓ2 ; z, z ′ ). ′ (C.1) ′ ′ We will prove that, for any z, z ∈ Λ such √ that z ′= z ′, and √ any w, w ∈ ′ Λa−1 ℓ1 ,a−1 ℓ2 such that w = w , w − z ≤ 2 and w − z  ≤ 2, 2 2 2gc (z, z ′ ) − agscal (ℓ1 , ℓ2 ; aw, aw′ )2 ≤ C(min{L, M, z − z ′ })−2 , (C.2) provided that min{L, M, z − z ′ } is sufficiently large. Proposition 2.9 readily follows from (C.2), simply by rescaling by a−1 . Note that, thanks to (C.1), ag ; aw, aw′ ) =√gscal (a−1 ℓ1 , a−1 ℓ2 ; w, w′ ); note also that |a−1 ℓ1 − L| ≤ √scal (ℓ1 , ℓ2−1ℓ 2 2 and |a 2 −M | ≤ 2. By using the explicit√expression of the scaling limit √ propagator (2.3.3) and the fact that w − z ≤ 2 and w′ − z ′  ≤ 2, we find that gscal (a−1 ℓ1 , a−1 ℓ2 ; w, w′ ) − gscal (L, M + 1; z, z ′ ) ≤ C(min{L, M, z − z ′ })−2 . Therefore, in order to prove (C.2), it is enough to show that, for min{L, M, z − z ′ } large, 2 2 2gc (z, z ′ ) − gscal (L, M + 1; z, z ′ )2 ≤ C(min{L, M, z − z ′ })−2 , (C.3) which is what we will prove )in the rest of this appendix. ∞ Recall that gc (z, z ′ ) = 0 g[η] (z, z ′ ) dη, with    1 [η] G (k1 , k2 ; z, z ′ ), (C.4) g[η] (z, z ′ ) = ♯ 2LNM (k1 , k2 ) ♯ ♯=± k1 ∈DL k2 ∈QM (k1 ) [η] G♯ where were defined in (B.12). Similarly, gscal (L, M + 1; z, z ′ ) = (L, M + 1; z, z ′ ) dη, with [η] gscal (L, M + 1; z, z ′ )   ♯ := ♯=±  π k1 ∈ L (2Z+1) k2 ∈ 2(Mπ+1) (2Z+1) )∞ 0 [η] gscal 1 [η] G (k1 , k2 ; z, z ′ ),(C.5) 2L(M + 1) scal;♯ where d(k1 , k2 ) := (1 − t2 )2 k12 + (1 − t1 )2 k22 and [η] ′ ′ [η] ′ ′ Gscal;+ (k1 , k2 ; z, z ′ ) :=e−ik1 (z1 −z1 )−ik2 (z2 −z2 )−ηd(k1 ,k2 )   −2it1 k1 −(1 − t21 )ik2 , −(1 − t21 )ik2 2it1 k1 Gscal;− (k1 , k2 ; z, z ′ ) :=e−ik1 (z1 −z1 )−ik2 (z2 +z2 )−ηd(k1 ,k2 )   −2it1 k1 (1 − t21 )ik2 . −(1 − t21 )ik2 e2ik2 (M +1) 2it1 k1 (C.6) We rewrite [η] [η] [η] gscal (L, M + 1; z, z ′ ) − g[η] (z, z ′ ) = R1 (z, z ′ ) + R2 (z, z ′ ), (C.7) 1130 G. Antinucci et al. where [η] R1 (z, z ′ )  := ♯=±  − with BL,M := Moreover, π L (2Z [η] R2 (z, z ′ ) 1 ♯ 2L(M + 1) "  Gscal;♯ (k1 , k2 ; z, z ′ ) (k1 ,k2 )∈BL,M π 2(M +1) (2Z  1  := ♯ L ♯=± − # [η] G♯ (k1 , k2 ; z, z ′ ) (k1 ,k2 )∈DL,M + 1) × Ann. Henri Poincaré k1 ∈DL  " k2 ∈QM (k1 ,k2 ) , (C.8) + 1), and DL,M := DL × D2(M +1) .  k2 ∈D2(M +1) 1 2(M + 1) # 1 [η] G (k1 , k2 ; z, z ′ ). 2NM (k1 , k2 ) ♯ (C.9) The first remainder term. We consider the contribution to gscal (L, M + [η] 1; z, z ′ ) − gc (z, z ′ ) from R1 first. Examining the definitions (B.12) and (C.6), [η] we see that, if (k1 , k2 ) ∈ DL,M , each matrix element of Gscal;♯ (k1 , k2 ; z, z ′ ) − [η] 2 G♯ (k1 , k2 ; z, z ′ ) is bounded in absolute value by C|k|2 e−cη|k| , with |k|2 = k12 + k22 , for some C, c > 0; if (k1 , k2 ) ∈ BL,M \ DL,M , each matrix ele2 [η] ment of Gscal;♯ (k1 , k2 ; z, z ′ ) is bounded in absolute value by C|k|e−cη|k| . [η] These bounds are sufficient for performing the integral of R1 (z, z ′ ) over η ≥ (min{L, M })2 . In fact, for such values of η, 1 2L(M + 1)  |k|2 |k| 2 (k1 ,k2 )∈BL,M e−cη|k| · if max{|k1 |, |k2 |} < π ≤ Cη −2 , otherwise (C.10) for some C > 0, so that  ∞  [η] ′ R1 (z, z )dη ≤ C ∞ (min{L,M })2 (min{L,M })2 η −2 dη ≤ C ′ (min{L, M })−2 . (C.11) [η] In order to bound the contribution from the integral of R1 (z, z ′ ) over [η] η ≤ (min{L, M })2 , we need to rewrite R1 (z, z ′ ) as a suitable integral in the complex plane, in analogy with what we did in Appendix B. More precisely, by using (B.20) and its analogue for the sums over k2 , we find that the matrix [η] elements of R1 (z, z ′ ) can be rewritten as:    [η] R1 (z, z ′ ) ωω′ = ♯ · ♯=± ·  ∞+iσ1 b −∞+iσ1 b dk1 2π  σ1 ,σ2 =0,± ∞+iσ2 b −∞+iσ2 b  dk2 ′ A (k1 ) A′′σ2 (k2 ) Gscal;♯ (k1 , k2 ; z, z ′ ) ωω′ 2π σ1 Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry −  π+iσ1 b dk1 2π −π+iσ1 b  π+iσ2 b −π+iσ2 b 1131 !  dk2 ′ [η] ′ ′′ A (k1 ) Aσ2 (k2 ) G♯ (k1 , k2 ; z, z ) ωω′ , 2π σ1 (C.12) where b will be conveniently fixed below, A′σ (k) was defined after (B.20) and A′′σ is defined analogously: A′′0 (k) ≡ 1 and, if σ = ±, A′′σ (k) = −e2iσk(M +1) /(1+ e2iσk(M +1) ). We now proceed as described before and after (B.23): if σ1 = 0, we shift k1 in the complex plane as k1 → k1 − ib sign(z1 − z1′ ); if σ2 = 0, depending on the values of ♯, ω, ω ′ , we shift k2 → k2 − iτ♯,(ωω′ ) b, with τ♯,(ωω′ ) as in (B.23). Next, we combine the third line of (C.12) with the contribution to the integral in the second line from the region max{|ℜk1 |, |ℜk2 |} ≤ π. After  [η] these manipulations, we find that R1 (z, z ′ ) ωω′ can be further rewritten as [η] R1 (z, z ′ )  ωω ′  =  ♯ σ1 ,σ2 =0,± ♯=±  ∞+iσ̃1 b −∞+iσ̃1 b dk1 2π  ∞+iσ̃2 b −∞+iσ̃2 b   [η] Gscal;♯ (k1 , k2 ; z, z ) ωω′ − G♯ (k1 , k2 ; z, z ′ ) ωω′  Gscal;♯ (k1 , k2 ; z, z ′ ) ωω′ · ′ dk2 ′ A (k1 ) A′′ σ2 (k2 ) · 2π σ1 if max{|ℜk1 |, |ℜk2 |} ≤ π if max{|ℜk1 |, |ℜk2 |} > π (C.13) where σ̃1 = −sign(z1 −z1′ ), if σ1 = 0, and σ̃1 = σ1 , otherwise; and σ̃2 = τ♯,(ωω′ ) , if σ2 = 0, and σ̃2 = σ2 , otherwise. We now pick b = η −1/2 and notice that, if ℑk1 = σ̃1 η −1/2 and ℑk2 = σ̃2 η −1/2 , with η ≤ (min{L, M })2 , the integrand in the right side of (C.12) is bounded in absolute value by Ceη −1/2 e−η −1/2 z−z ′ −cη|k|2 |k|2 + η −1 |k| + η −1/2 · if max{|ℜk1 |, |ℜk2 |} ≤ π if max{|ℜk1 |, |ℜk2 |} > π for some C, c > 0. Therefore, recalling that z − z ′  ≫ 1, [η] R1 (z, z ′ ) + − 21 η −1/2 z−z ′  ≤ Ce  "  2 dk e−cη|k| (|k|2 + η −1 ) [−π,π]2 2 # 1 dk e−cη|k| (|k| + η −1/2 ) ≤ C ′ e− 2 η R2 \[−π,π]2 −1/2 z−z ′  ′ (η −2 +e−c η η −3/2 ). (C.14) 1 −1/2 ′ ′ z−z −c η Note that e− 4 η ≤ Ce−c 2 η ≤ (min{L, M }) , we find:  0 (min{L,M })2 [η] ′′ z−z ′ 2/3 , so that, by integrating over R1 (z, z ′ )dη ≤ C(z − z ′ −2 + z − z ′ −1 e−c ≤ C ′ z − z ′ −2 . ′′ z−z ′ 2/3 ) (C.15) 1132 G. Antinucci et al. Combining this with (C.11), we find that z ′ })−2 . )∞ 0 Ann. Henri Poincaré [η] R1 (z, z ′ )dη ≤ C(min{L, M, z− The second remainder term. Let us now consider the contribution to [η] gscal (L, M + 1; z, z ′ ) − gc (z, z ′ ) from R2 . By using (B.18) and the analogue of [η] (B.20) for the sums over k2 in D2(M +1) , we rewrite R2 (z, z ′ ) as  1    π+iσb dk2 [η] R2 (z, z ′ ) := ♯ L 2π ♯=± k1 ∈DL σ=± −π+iσb  [η] × A′′σ (k2 ) − Aσ (k1 , k2 ) G♯ (k1 , k2 ; z, z ′ ), (C.16) where A′′σ and Aσ were defined after (C.12) and in (B.19), respectively. Note that, for σ = ±, A′′σ (k2 ) − Aσ (k1 , k2 ) = − =− e2iσk(M +1) Rσ (k1 , k2 )e2iσk2 (M +1) − 2iσk(M +1) 1+e 1 − Rσ (k1 , k2 )e2iσk2 (M +1) 2(1 + Rσ (k1 , k2 ))e2iσk2 (M +1) . (1 + e2iσk2 (M +1) )(1 − Rσ (k1 , k2 )e2iσk2 (M +1) ) (C.17) Recalling the definition (B.17) of Rσ , we have   2 2 2  1 − B(k1 ) cos(k2 )   ≤ C |k1 | + |k2 | + b |1 + Rσ (k1 , k2 )| = 2   iσk 1 − B(k1 )e 2 b (C.18) for |ℑk1 | ≤ σℑk2 = b positive and sufficiently small; and recalling the bound (B.26) on Rσ , which remains valid with iσ̃b replaced by iσ̃b′ , |b′ | ≤ b, the denominator of (C.17) is bounded from below as |1 − Rσ (k1 , k2 )e2iσk2 (M +1) ||1 + e2iσk2 (M +1) | ≥ (1 − e−b(M +1)) 2 (C.19) for M larger than some constant and |ℑk1 | ≤ σℑk2 = b positive and sufficiently small. We now proceed slightly differently, depending on whether η is larger or smaller than (ℓ(z, z ′ ))2 , with ℓ(z, z ′ ) := max{min{L, M }, z − z ′ }. The case of η smaller than (ℓ(z, z ′ ))2 . In this case, we rewrite (C.16) by using (B.20); if σ ′ = 0, we perform the complex shift k1 → k1 − ib sign(z1 − z1′ ), thus getting (letting σ̃ ′ = σ ′ , if σ ′ = ±, and σ̃ ′ = −sign(z1 − z1′ ), if σ ′ = 0)     π+iσ̃′ b dk1  π+iσb dk2 [η] A′σ′ (k1 ) ♯ R2 (z, z ′ ) := 2π ′ b 2π −π+iσ̃ −π+iσb ♯=± σ ′ =0,± σ=±  [η] A′′σ (k2 ) − Aσ (k1 , k2 ) G♯ (k1 , k2 ; z, z ′ ). (C.20) We now bound the integrand by its absolute value, by using, in particular, [η] (C.17)–(C.18), and by estimating the matrix elements of G♯ in the same way as we did several times above and in Appendix B. We thus get, for b = Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry 1133 c0 min{1, η −1/2 } with c0 sufficiently small,  ′ 2 2 Ce−bz−z  (|k| + b)3 −η|k|2 2 [η] ′ 2 e dk 2R2 (z, z )2 ≤ (1 − e−bL )(1 − e−bM )2 [−π,π]2 b ′ Cb4 e−bz−z  ≤ . (1 − e−bL )(1 − e−bM )2 (C.21) If we now integrate this inequality with respect to η, for 0 ≤ η ≤ ℓ(z, z ′ ), recalling that ℓ(z, z ′ ) = max{min{L, M }, z − z ′ }, we find, for z − z ′ 1,  1  (ℓ(z,z′ ))2 2 2 ′ 2 [η] ′ 2 e−c0 z−z  dη 2R2 (z, z )2 dη ≤ C 0 0 +C  (min{L,M })2 η −2 e−c0 η −1/2 z−z ′  dη 1 +1(z − z ′  > min{L, M })  z−z′ 2 −1/2 C z−z ′  η −1/2 e−c0 η dη LM 2 (min{L,M })2  ′ 1 z − z ′   ≤ C ′ e−c0 z−z  + + z − z ′ 2 LM 2 ′′ C ≤ . (C.22) (min{L, M, z − z ′ })2 The case of η larger than ℓ(z, z ′ ). In this case we go back to the representation (C.16) (no rewriting of the sum over k1 in terms of an integral in the complex plane). We proceed slightly differently for the diagonal and off-diagonal elements [η] of R2 . Let us begin with the diagonal terms. Note that the diagonal elements [η] of G♯ have the form ′ ±2it1 e−ik1 (z1 −z1 ) e−ik2 Z2 e−ηD(k1 ,k2 ) sin k1 where Z2 is either z2 − z2′ , z2 + z2′ , or z2 + z2′ − 2M − 2. We thus see that each [η] diagonal element of R2 is given by a sum of four terms (due to the sums over ♯ and σ) of the form ± 2it1 L   π+iσb −π+iσb k1 ∈DL  −ik (z −z′ ) −ik Z −ηD(k ,k ) 1 1 2 2 1 2 1 e A′′ e σ (k2 ) − Aσ (k1 , k2 ) e dk2 2π (C.23) ′  π+iσb 2(1 + Rσ (k1 , k2 ))eiσk2 Z2,σ sin k1 (z1 − z1′ )e−ηD(k1 ,k2 ) sin k1 2t1  =± L k ∈D −π+iσb (1 − Rσ (k1 , k2 )e2iσk2 (M +1) )(1 + e2iσk2 (M +1) ) sin k1 1 L dk2 , 2π where in passing from the first to the second line we used (C.17) and the ′ fact that Rσ (k1 , k2 ) is even in k1 . Moreover, in the second line, Z2,σ is either ′ ′ 2(M + 1) − σ(z2 − z2 ), 2(M + 1) − σ(z2 + z2 ), or 2(M + 1)(1 + σ) − σ(z2 + z2′ ); 1134 G. Antinucci et al. Ann. Henri Poincaré ′ in any case, Z2,σ ≥ 2. We can then use this, together with Inequalities (C.18) and (C.19) and the observation that | sin k1 (z1 − z1′ )| ≤ |k1 | · z − z ′  to obtain, for ω = ± and b = c0 η −1/2 , with η ≥ ℓ(z, z ′ ),    C z − z′  [η]  R2 (z, z ′ ) ωω  ≤ L ′ ≤   k1 ∈DL C z−z η M2 π dk2 −π  η 1/2 k12 + k22 + η −1 2 2 |k1 |2 e−cη(k1 +k2 ) (1 − e−c0 η−1/2 M )2 −3/2 (C.24) , and thus, recalling that ℓ(z, z ′ ) = max{min{L, M }, z − z ′ },     ∞  Cz − z ′  C   [η] ′ R2 (z, z ) ωω dη  ≤ 2 ≤ 2,   (ℓ(z,z′ ))2  M ℓ(x, y) M (C.25) which is of the desired order. [η] The off-diagonal elements of G♯ are equal, up to a sign, to ′ 2(1 − t21 )e−ik1 (z1 −z1 ) e−ik2 Z2 e−ηD(k1 ,k2 ) (1 − B(k1 )e±ik2 ), where as before Z2 is either z2 − z2′ , z2 + z2′ , or z2 + z2′ − 2M − 2. Noting that R± (k1 , k2 ) = R∓ (k1 , −k2 ), we rewrite  π−ib  [η] dk2 A′′− (k2 ) − A− (k1 , k2 ) G♯ (k1 , k2 ; z, z ′ ) 2π −π−ib  π+ib  [η] dk2 (C.26) A′′+ (k2 ) − A+ (k1 , k2 ) G♯ (k1 , −k2 ; z, z ′ ) = 2π −π+ib [η] and so each off-diagonal element of R2 can be written as a sum of two terms (due to the sum over ♯) of the form  π+ib ′ 1  2(1 + R+ (k1 , k2 ))e2ik2 (M +1) e−ik1 (z1 −z1 ) e−ηD(k1 ,k2 ) L k ∈D −π+ib (1 − R+ (k1 , k2 )e2ik2 (M +1) )(1 + e2ik2 (M +1) ) 1 L × e−ik2 Z2 (1 − B(k1 )e±ik2 ) + eik2 Z2 (1 − B(k1 )e∓ik2 ) up to uninteresting coefficients; then noting that 1 2  dk2 2π  e−ik2 Z2 (1 − B(k1 )e±ik2 ) + eik2 Z2 (1 − B(k1 )e∓ik2 ) = cos k2 Z2 − B(k1 ) cos k2 (Z2 ∓ 1) = (1 − cos k2 ) cos k2 Z2 ∓ sin k2 sin k2 Z2 + [1 − B(k1 )] cos k2 (Z2 ∓ 1), we obtain, for ω = ± and b = c0 η −1/2 with η ≥ ℓ(z, z ′ ), noting also that |Z2 | ≤ 2M ,  2  C|Z |   π   η 1/2 k12 + k22 + η −1 2 2   [η] 2 ′ e−cη(k1 +k2 ) dk2  R2 (z, z ) ω,−ω  ≤ −1/2 M 2 −c η 0 L (1 − e ) −π k ∈D 1 C −3/2 η , ≤ M L (C.27) Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry from which 1135    ∞    C   [η] ′ , R2 (z, z ) dη  ≤   (ℓ(z,z′ ))2  M ℓ(z, z ′ ) ω,−ω (C.28) which is again of the desired order. Combining this with (C.25) and (C.22), 2 ) ∞ 2 [η] we find that 0 2R2 (z, z ′ )2 dη ≤ C(min{L, M, z − z ′ })−2 . Together with [η] the bound on R1 , see the line after (C.15), this concludes the proof of (C.3) and, therefore, of Proposition 2.9. D. Non-interacting Correlation Functions in the Scaling Limit In this appendix, we explain how to express the scaling limit of the noninteracting correlation function appearing in Theorem 1.1 in terms of the propagators studied in Appendix C, thus proving, in particular, [4, Eq.(1.12)]. For notational simplicity, in this appendix we let t∗1 (λ) ≡ t1 and t∗2 (λ) ≡ t2 = (1 − t1 )/(1 + t1 ). By using the Grassmann representation of Proposition 3.1 in the case λ = 0, we find that, for the lattice of unit mesh and any m-tuple of distinct edges x1 , . . . , xm , with m ≥ 2, ǫx1 ; · · · ; ǫxm 0,t1 ,t2 ;Λ = ∂m log ∂Ax1 · · · ∂Axm  St1 ,t2 (Φ)+ DΦe  2 x∈BΛ (1−tj(x) )Ex Ax    A=0 (D.1) (note that the expectation in the left side is the truncated one). Introducing the rescaled energy observable εaℓ (z) := a−1 σz σz+aêℓ , rescaling the lattice by a factor of a and passing over to the non-truncated expectation, we obtain  a  εl1 (z1 ) · · · εalm (zm ) 0,t ,t ;Λa 1 2      = a−m (1 − t21 )m1 (1 − t22 )m2 Ex(z1 ,l1 ) − Ex(z1 ,l1 ) · · · Ex(zm ,lm )   − Ex(zm ,lm ) , (D.2) where, in the right side: Ex(z,1) = H z Hz+aê1 and Ex(z,2) = V z Vz+aê2 ; the symbol (·) indicates normalized Grassmann measure Sta1 ,t2 a ) Sa (Φ) DΦe t1 ,t2 (·) ) Sa (Φ) , DΦe t1 ,t2 with the same as (2.1.2) on the rescaled lattice Λ . Recall the transformation (2.1.10) relating the variables {H z , Hz , V z , ω∈{±} Vz }z∈Λ to {φω,z , ξω,z }z∈Λ , from which we see that, if x is a vertical edge of endpoints z, z + aê2 , then Ex = φ+,z φ−,z+aê2 , while, if x is a horizontal edge of endpoints z, z + aê1 , then   (with obvious notation) Ex = s+ ∗(φ+ −φ− )(z) s− ∗(φ+ +φ− )(z +aê1 ) plus terms involving the ‘massive’ variables {ξω,z }z∈Λ,ω∈{±} . The reader can convince herself that, for the purpose of computing the limit a → 0+ of (D.2), in the right side of (D.2) we can freely replace Ex by the following local expressions in the Grassmann ‘massless’ variables: φ+,z φ−,z , if x is a vertical edge of endpoints z, z + aê2 (note that φ+,z φ−,z is obtained from φ+,z φ−,z+aê2 by ‘localizing’ the second field at the same position of the first 1136 G. Antinucci et al. Ann. Henri Poincaré one); and (1 + t1 )−2 (φ+,z − φ−,z )(φ+,z + φ−,z ) = 2(1 + t1 )−2 φ+,z φ−,z , if x is a vertical edge of endpoints z, z + aê1 (note that (1 + t1 )−2 (φ+,z − φ−,z  )(φ+,z + φ−,z ) is obtained from s+ ∗ (φ+− φ− )(z) s− ∗ (φ+ + φ− )(z + aê1 ) by localizing s− ∗ (φ+ + φ− )(z + aê1 ) at z, and by replacing the non-local, exponentially decaying, kernels s± (z1 ) by their local counterparts, namely c0 δz1 ,0 , L with c0 = limL→∞ y=1 s± (y) = (1 + t1 )−1 ). It is, in fact, easy to check that the difference between the exact expression of Ex and such a ‘local approximations’ is of higher order in a and its contribution to the correlation function vanishes in the limit a → 0. Therefore,   lim+ εal1 (z1 ) · · · εalm (zm ) 0,t1 ,t2 ;Λa a→0 = lim+ a−m a→0 2(1 − t21 ) (1 + t1 )2 m1 (1 − t22 )m2 : φ+,z1 φ−,z1 : · · · : φ+,zm φ−,zm :, (D.3) where : φ+,z φ−,z : denotes the difference φ+,z φ−,z − φ+,z φ−,z . Note that 2(1−t21 ) (1+t1 )2 = 2t2 . The Grassmann average in the right side of (D.3) can be expressed in terms of the fermionic Wick rule or, equivalently, in terms of the Pfaffian of the 2m × 2m anti-symmetric matrix Ma (z), whose elements, labeled by the indices (1, +), (1, −), . . . , (m, +), (m, −), are equal to    φω,zi φω′ ,zj if i = j, a M (z) (i,ω)(j,ω′ ) = 0 otherwise.  In view of Proposition 2.9, lima→0 a−1 φω,z φω′ ,z′  = gscal (z, z ′ ) ωω′ and, therefore,   lim εal1 (z1 ) · · · εalm (zm ) 0,t ,t ;Λa = (2t2 )m1 (1 − t22 )m2 Pf(M(z)), (D.4) a→0+ 1 2 with  M(z) (i,ω),(j,ω′ ) =  gscal (zi , zj ) ωω′ 0 if i = j, otherwise, (D.5) Since gscal is covariant under rescaling, see (C.1), the scaling limit (D.4) is, as well. Note that rescalings are, together with translations and parity, the only conformal transformations from finite cylinders to finite cylinders or, equivalently, from a finite circular annulus to a finite circular annulus: in fact, it is well known [5,28] that an annulus {z ∈ C : r < |z| < R} can be conformally mapped to another one only if the two annuli have 1 log(R/r); moreover, any automorphism of the annulus the same modulus 2π {z ∈ C : r < |z| < R} is either a rotation z → zeiθ or a rotation followed by an inversion z → Rr/z. 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Journal für die reine und angewandte Mathematik 83, 300–351 (1877) [29] Smirnov, S.: Conformal invariance in random cluster models. I. Holmorphic fermions in the Ising model. Ann. Math. 172, 1435–1467 (2010) Giovanni Antinucci Section de mathématiques Université de Genève 2-4 rue du Lièvre, 1211 Genève 4 Geneva Switzerland Alessandro Giuliani and Rafael L. Greenblatt Dipartimento di Matematica e Fisica Università degli Studi Roma Tre L.go S. L. Murialdo 1 00146 Roma Italy Alessandro Giuliani Centro Linceo Interdisciplinare Beniamino Segre Accademia Nazionale dei Lincei, Palazzo Corsini Via della Lungara 10 00165 Roma Italy e-mail: giuliani@mat.uniroma3.it Vol. 23 (2022) Non-integrable Ising Models in Cylindrical Geometry Present Address Rafael L. Greenblatt Scuola Internazionale Superiore di Studi Avanzati (SISSA) Mathematics Area, Via Bonomea 265 34136 Trieste Italy Communicated by Vieri Mastropietro. Received: April 21, 2021. Accepted: August 23, 2021. 1139