We extend proofs of non-Gibbsianness of decimated Gibbs measures at low temperatures to include l... more We extend proofs of non-Gibbsianness of decimated Gibbs measures at low temperatures to include long-range as well as vector-spin interactions. Our main tools consist in a two-dimensional use of “equivalence of boundary conditions” in the long-range case and an extension of global specifications for two-dimensional vector spins.
We discuss what ground states for generic interactions look like. We note that a recent result, d... more We discuss what ground states for generic interactions look like. We note that a recent result, due to Morris, implies that the behaviour of ground-state measures for generic interactions is similar to that of generic measures. In particular, it follows from his observation that they have singular spectrum and that they are weak mixing, but not mixing.
We study a variant of the ferromagnetic Potts model, recently introduced by Tamura, Tanaka and Ka... more We study a variant of the ferromagnetic Potts model, recently introduced by Tamura, Tanaka and Kawashima, consisting of a ferromagnetic interaction among q “visible” colours along with the presence of r non-interacting “invisible” colours. We introduce a random-cluster representation for the model, for which we prove the existence of a first-order transition for any q > 0, as long as r is large enough. When q > 1, the low-temperature regime displays a q-fold symmetry breaking. The proof involves a Pirogov-Sinai analysis applied to this random-cluster representation of the model.
We consider one-dimensional long-range spin models (usually called Dyson models), consisting of I... more We consider one-dimensional long-range spin models (usually called Dyson models), consisting of Ising ferromagnets with slowly decaying long-range pair potentials of the form \( {1}/{| i-j |^\alpha }\), mainly focusing on the range of slow decays \(1 < \alpha \le 2\). We describe two recent results, one about renormalization and one about the effect of external fields at low temperature.
We consider ferromagnetic long-range Ising models which display phase transitions. They are long-... more We consider ferromagnetic long-range Ising models which display phase transitions. They are long-range one-dimensional Ising ferromagnets, in which the interaction is given by J_x,y = J(|x-y|)≡1/|x-y|^2-α with α∈ [0, 1), in particular, J(1)=1. For this class of models one way in which one can prove the phase transition is via a kind of Peierls contour argument, using the adaptation of the Fröhlich-Spencer contours for α≠ 0, proposed by Cassandro, Ferrari, Merola and Presutti. As proved by Fröhlich and Spencer for α=0 and conjectured by Cassandro et al for the region they could treat, α∈ (0,α_+) for α_+=(3)/(2)-1, although in the literature dealing with contour methods for these models it is generally assumed that J(1)≫1, we can show that this condition can be removed in the contour analysis. In addition, combining our theorem with a recent result of Littin and Picco we prove the persistence of the contour proof of the phase transition for any α∈ [0,1). Moreover, we show that when we...
We extend proofs of non-Gibbsianness of decimated Gibbs measures at low temperatures to include l... more We extend proofs of non-Gibbsianness of decimated Gibbs measures at low temperatures to include long-range as well as vector-spin interactions. Our main tools consist in a two-dimensional use of “equivalence of boundary conditions” in the long-range case and an extension of global specifications for two-dimensional vector spins.
We discuss what ground states for generic interactions look like. We note that a recent result, d... more We discuss what ground states for generic interactions look like. We note that a recent result, due to Morris, implies that the behaviour of ground-state measures for generic interactions is similar to that of generic measures. In particular, it follows from his observation that they have singular spectrum and that they are weak mixing, but not mixing.
We study a variant of the ferromagnetic Potts model, recently introduced by Tamura, Tanaka and Ka... more We study a variant of the ferromagnetic Potts model, recently introduced by Tamura, Tanaka and Kawashima, consisting of a ferromagnetic interaction among q “visible” colours along with the presence of r non-interacting “invisible” colours. We introduce a random-cluster representation for the model, for which we prove the existence of a first-order transition for any q > 0, as long as r is large enough. When q > 1, the low-temperature regime displays a q-fold symmetry breaking. The proof involves a Pirogov-Sinai analysis applied to this random-cluster representation of the model.
We consider one-dimensional long-range spin models (usually called Dyson models), consisting of I... more We consider one-dimensional long-range spin models (usually called Dyson models), consisting of Ising ferromagnets with slowly decaying long-range pair potentials of the form \( {1}/{| i-j |^\alpha }\), mainly focusing on the range of slow decays \(1 < \alpha \le 2\). We describe two recent results, one about renormalization and one about the effect of external fields at low temperature.
We consider ferromagnetic long-range Ising models which display phase transitions. They are long-... more We consider ferromagnetic long-range Ising models which display phase transitions. They are long-range one-dimensional Ising ferromagnets, in which the interaction is given by J_x,y = J(|x-y|)≡1/|x-y|^2-α with α∈ [0, 1), in particular, J(1)=1. For this class of models one way in which one can prove the phase transition is via a kind of Peierls contour argument, using the adaptation of the Fröhlich-Spencer contours for α≠ 0, proposed by Cassandro, Ferrari, Merola and Presutti. As proved by Fröhlich and Spencer for α=0 and conjectured by Cassandro et al for the region they could treat, α∈ (0,α_+) for α_+=(3)/(2)-1, although in the literature dealing with contour methods for these models it is generally assumed that J(1)≫1, we can show that this condition can be removed in the contour analysis. In addition, combining our theorem with a recent result of Littin and Picco we prove the persistence of the contour proof of the phase transition for any α∈ [0,1). Moreover, we show that when we...
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