Ann. Probab. 29 (2001), 1–65
Version of 1 June 2005
Uniform Spanning Forests
by Itai Benjamini, Russell Lyons, Yuval Peres, and Oded Schramm
Abstract. We study uniform spanning forest measures on infinite graphs,
which are weak limits of uniform spanning tree measures from finite subgraphs.
These limits can be taken with free (FSF) or wired (WSF) boundary conditions.
Pemantle (1991) proved that the free and wired spanning forests coincide in
Zd and that they give a single tree iff d 6 4.
In the present work, we extend Pemantle’s alternative to general graphs
and exhibit further connections of uniform spanning forests to random walks,
potential theory, invariant percolation, and amenability. The uniform spanning forest model is related to random cluster models in statistical physics,
but, because of the preceding connections, its analysis can be carried further.
Among our results are the following:
• The FSF and WSF in a graph G coincide iff all harmonic Dirichlet
functions on G are constant.
• The tail σ-fields of the WSF and the FSF are trivial on any graph.
• On any Cayley graph that is not a finite extension of Z, all component
trees of the WSF have one end; this is new in Zd for d > 5.
• On any tree, as well as on any graph with spectral radius less than 1,
a.s. all components of the WSF are recurrent.
• The basic topology of the free and the wired uniform spanning forest
measures on lattices in hyperbolic space Hd is analyzed.
• A Cayley graph is amenable iff for all ǫ > 0, the union of the WSF and
Bernoulli percolation with parameter ǫ is connected.
• Harmonic measure from infinity is shown to exist on any recurrent
proper planar graph with finite co-degrees.
We also present numerous open problems and conjectures.
1991 Mathematics Subject Classification. Primary 60D05. Secondary 05C05, 60B99, 20F32, 31C20, 05C80.
Key words and phrases. Spanning trees, Cayley graphs, electrical networks, harmonic Dirichlet functions,
amenability, percolation, loop-erased walk.
Research partially supported by the Institute for Advanced Studies, Jerusalem and a Varon Visiting
Professorship at the Weizmann Institute of Science (Lyons), NSF grant DMS-94-04391 (Peres), and the
Sam and Ayala Zacks Professorial Chair (Schramm).
1
§1. Introduction
2
Contents
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Introduction
Basic Definitions
Wilson’s Method
Electrical Networks and Random Spanning Trees
Basic Properties of Random Spanning Forests
Average and Expected Degrees
Potential Theory
Ergodic Properties
The Number of Components
Ends of WSF Components in Transitive Graphs
Analysis of the WSF on a Tree
Planar Graphs and Hyperbolic Lattices
The WSF in Nonamenable Graphs
Applications to Loop-Erased Walks and Harmonic Measure
Open Questions
References
2
5
7
9
16
20
22
26
32
35
46
51
55
63
65
68
§1. Introduction.
Combinatorialists have long known that much information about a finite graph is encoded
in its ensemble of spanning trees. A beautiful illustration of this was the algorithm found
independently by Aldous (1990) and Broder (1989) for generating uniformly a random
spanning tree using simple random walk. By analogy with Gibbs measures in statistical
mechanics, one might expect that limits of uniform spanning tree measures could be constructed on infinite graphs, and that boundary conditions and the dimensionality would be
important. Indeed, motivated by some questions of R. Lyons, Pemantle (1991) showed that
if an infinite graph G is exhausted by finite subgraphs Gn , then the uniform distributions
on the spanning trees of Gn converge weakly to a measure supported on spanning forests*
of G. We call this the free uniform spanning forest (FSF), since there is another natural construction where the exterior of Gn is identified to a single vertex (“wired”) before
passing to the limit. This second construction, which we call the wired uniform spanning forest (WSF), was implicit in Pemantle’s paper and was made explicit by Häggström
(1995).
Pemantle (1991) discovered the following interesting properties, among others:
• The free and the wired uniform spanning forest measures are the same on all eu-
clidean lattices Zd . This implies that they have trivial tail σ-fields.
* By a “spanning forest”, we mean a subgraph without cycles that contains every vertex.
§1. Introduction
3
• On Zd , the uniform spanning forest is a single tree a.s. if d 6 4; but when d > 5,
there are infinitely many trees a.s.
• If 2 6 d 6 4, then the uniform spanning tree on Zd has a single end a.s. (as defined
in Section 2); when d > 5, each of the infinitely many trees a.s. has at most two ends.
One of Pemantle’s main tools was the Aldous-Broder algorithm. In addition, Lawler’s deep
analysis of loop-erased random walks in Zd was crucial.
In the present work, we redevelop the fundamentals of the theory, broaden its scope,
and present new results. We believe that the area of uniform spanning forests presents a
rich object of study. It has important connections to several areas, such as random walks,
algorithms, domino tilings, electrical networks, potential theory, amenability, percolation,
and hyperbolic spaces. We expect significant connections to conformal mapping and to
(continuous) stochastic processes. Because of this, there are still many fascinating open
questions and conjectures to settle; we anticipate much further work in this field.
Among our results are the following:
• A geometric proof is given (Section 4) of the Transfer Current Theorem of Burton
and Pemantle (1993).
• Wilson’s (1996) algorithm is adapted to infinite graphs (Theorem 5.1).
• We show that the free and the wired uniform spanning forest measures are the same
iff the graph does not support any nonconstant harmonic Dirichlet functions (Theorem 7.3).
• We prove that the free and the wired uniform spanning forest measures have trivial
tail σ-fields on every graph (Theorem 8.3), and give a quantitative estimate for correlations
of cylinder events (Theorem 8.4).
• The number of trees of the wired uniform spanning forest on every graph is determined (Theorems 9.2 and 9.4).
• We complete and extend Pemantle’s (1991) determination of the number of ends
by showing that for the wired uniform spanning forest on any Cayley graph that is not a
finite extension of Z, each tree has one end a.s. (Theorem 10.1).
• We prove that on any tree, as well as on any graph with spectral radius less than
1, a.s. all components of the WSF are recurrent (Theorem 11.1 and Corollary 13.4).
• We show that on any proper planar recurrent graph that has a finite number of
sides to each face, the uniform spanning forest (which is a tree) has only one end a.s.
(Theorem 12.4).
• The basic topology of the free and the wired uniform spanning forest measures on
lattices in hyperbolic space Hd is analyzed (Theorem 12.7).
• We prove that a Cayley graph G is amenable iff for all ǫ > 0, the wired uniform
§1. Introduction
4
spanning forest on G a.s. becomes connected when edges are added independently with
probability ǫ each (Theorem 13.7 and the discussion above it).
• Harmonic measure from infinity is shown to exist on any proper planar recurrent
graph that has a finite number of sides to each face (Theorem 14.2).
Our results are based on several recently developed tools. The most important is an
algorithm invented by Wilson (1996) to generate random spanning trees of finite graphs;
extending it to infinite graphs allows us to generate the WSF directly, without weak limits.
The second tool is a “mass-transport principle” that was developed in the context of
group-invariant percolation (Häggström (1997) and Benjamini, Lyons, Peres, and Schramm
(1999), denoted BLPS (1999) below); and the third is a general property of loop-erased
Markov chains, established in Lyons, Peres, and Schramm (1998).
Häggström (1995) showed that the uniform spanning forest measures in Z d arise as
limits of (Fortuin-Kasteleyn) random cluster measures. Such measures generalize ordinary
(Bernoulli) percolation, Ising and Potts models of statistical physics (see Grimmett (1995)
for a review). Many questions that are difficult and often still unsolved become tractable
for the uniform spanning forest model because of its close connection to potential theory
and random walks. (For example, it is not known precisely when limits of random cluster
measures with free and wired boundary conditions coincide.)
Although the uniform measure on spanning trees is the most often used, there are other
natural measures as well. The general context in which we shall work is that in which every
edge is given a weight and a spanning tree is chosen with probability proportional to the
product of the weights of its edges. A natural setting in which nonuniform weights are
interesting is that of a Cayley graph in which different generators get different weights.
Fortunately, the extra generality presents no additional significant difficulty. We shall use
the notations FSF and WSF for the general case, as well as the uniform case, of free and
wired spanning forest measures.
Besides Cayley graphs and (vertex-) transitive graphs, other especially interesting
classes of graphs on which we analyze the spanning forests are trees (Section 11), planar
graphs and hyperbolic lattices (Section 12), and nonamenable graphs (Section 13).
In Sections 2–4, we give a self-contained and rapid development of the theory of spanning trees on finite graphs, except for the proof of the correctness of Wilson’s algorithm.
For a more leisurely development, one may consult Lyons (1998). In Sections 4 and 7,
we develop the relations to electric networks, random walks and potential theory using
a purely Hilbert-space approach. Although these results and this approach are classical,
we present them in a particularly transparent manner, preferring geometric over cohomo-
§2. Basic Definitions
5
logical terminology. Here, “geometry” refers both to the graph and to the Hilbert space.
The interaction of these two geometries is explored more fully in our new proof of the
Transfer Current Theorem (Section 4) and is developed more deeply in our quantitative
proof of tail triviality (Section 8). In amenable graphs, such as Z d , there is a natural way
of averaging; it leads immediately in Section 6 to the fact that the average degree in any
spanning forest of infinite trees is two. In particular, the free and wired uniform spanning
forests agree on a transitive amenable graph and give each vertex expected degree two.
The number of trees in the wired spanning forest is determined in Section 9; the case of
the free spanning forest is still largely mysterious. That each tree in the wired uniform
spanning forest on a Cayley graph has only one end a.s. (except for finite extensions of
Z) is proved in Section 10. An application of the study of spanning forests to harmonic
Dirichlet functions is given in Section 7; more applications appear in Benjamini, Lyons
and Schramm (1999). Applications to loop-erased random walk and harmonic measure
from infinity are in Section 14. Information on the “size” of the trees in the wired uniform
spanning forest on nonamenable graphs, as well as on their connectivity when edges are
added randomly and independently, appears in Section 13.
A collection of open questions is presented in Section 15. For example, conformal
invariance conjectures suggest that the uniform spanning tree in Z2 and other lattices of
R2 should be investigated further: see Question 15.13.
Another model of random spanning forests that is closely connected to Bernoulli
percolation is the minimal spanning forest (see, e.g., Alexander (1995)). There are several
parallels between the minimal spanning forest and the uniform spanning forest; we intend
to develop some of these in a future publication.
§2. Basic Definitions.
A forest is a graph with no cycles. A tree is a nonempty connected forest. A
subgraph H ⊆ G is spanning if H contains all the vertices of G. We shall be interested in
spanning forests and spanning trees. A spanning tree or forest of G = (V, E) will usually
be thought of as a subset of E.
In an undirected graph, a spanning tree is composed of undirected edges. However,
we shall often consider flows on graphs, and therefore use directed edges as well. Multiple
edges joining the same two vertices, as well as loops (edges joining a vertex to itself), are
allowed in a graph, but note that loops are never contained in a forest.
For a graph G = (V, E) with vertex set V and (directed) edge set E, write e, e ∈ V for
the tail and head of e ∈ E; the edge is oriented from its tail to its head. Write ě for the
§2. Basic Definitions
6
reverse orientation. Each edge occurs with both orientations.
A network is a pair (G, C), where G is a connected graph with at least two vertices
and C is a function from the unoriented edges of G to the positive reals. Often, we
shall omit mention of C, and just call the network G. The quantity C(e) is called the
conductance of the edge e. The network is finite if G is finite. The conductance of
P
a vertex v in the network is Cv :=
{C(e) : e = v}. We generally assume in the
following that the networks under discussion satisfy Cv < ∞ for all v ∈ V. If additionally
supv∈V Cv < ∞, we say that the network has bounded vertex conductance. The most
natural network on a graph G is the default network (G, 1). For every edge e ∈ E, we call
R(e) := 1/C(e) the resistance of e.
Given a network (G, C) and a vertex v ∈ V, there is an associated Markov chain
hX(0), X(1), . . .i on V with distribution Pv . It has initial state X(0) = v and transition
probabilities
Pv [X(n + 1) = w | X(n) = u] = C(u, w)/Cu ,
where
C(u, w) :=
X
{C(e) : e = u, e = w} .
This Markov chain is called the network random walk starting at v. When there is a
need to indicate the starting vertex v, the notation Xv is often used. (If there is more
than one edge joining a pair of vertices, it is sometimes important to record not only the
sequence of vertices visited by this Markov chain, but also the edges used. When X(n) = u,
the probability that X will use the edge e satisfying e = u at the next step is C(e)/C u .)
Of course, the class of network random walks is the same as the class of reversible Markov
chains.
Given a random walk hX(n)i, we use the following notations for hitting times:
τA := inf{n > 0 : X(n) ∈ A} ,
τA+ := inf{n > 0 : X(n) ∈ A} .
+
Similarly, τv := τ{v} and τv+ := τ{v}
are the hitting times of a vertex v.
A graph automorphism ϕ of a graph G = (V, E) is a pair of bijections ϕV : V → V
and ϕE : E → E such that ϕV maps the tail and head of e to the tail and head, respectively,
of ϕE (e). A network automorphism ϕ : (G, C) → (G, C) is a graph automorphism of
G such that C(ϕ(e)) = C(e) for all e ∈ E. A network or graph G is transitive if for every
v, u ∈ V, there is an automorphism of G taking v to u. The group of all automorphisms
of G will be denoted by Aut(G).
§3. Wilson’s Method
7
Given any graph G = (V, E), we let 2E denote the measurable space of all subsets of E
with the Borel σ-field, that is, the σ-field generated by sets of the form {F ⊆ E : e ∈ F },
where e ∈ E. An elementary cylinder is an event A ⊆ 2E of the form A = {F ∈ 2E :
F ∩ K = B}, where K, B ⊆ E are finite. A cylinder event is a finite union of elementary
cylinders.
An infinite path in a tree that starts at any vertex and does not backtrack is called
a ray. Two rays are equivalent if they have infinitely many vertices in common. An
equivalence class of rays is called an end. In a graph G that is not a tree, the notion
of end is slightly harder to define. An end of G is a mapping ξ that assigns to any
finite set K ⊂ V an infinite component of G \ K and satisfies the consistency condition
K0 ⊂ K =⇒ ξ(K0 ) ⊃ ξ(K). It is easy to verify that when G is a tree, these two definitions
of an end are equivalent.
§3. Wilson’s Method.
Let G be a finite connected network; later, we shall generalize the discussion to infinite
networks.
Every connected graph has a spanning tree. In fact, the number of spanning trees is
“typically” exponential in the size of the graph. Thus, it is not obvious how to choose
one uniformly at random in polynomial time, and several sophisticated algorithms have
been devised to do this. The “fastest” algorithm known, which we describe below, is due
to Wilson (1996). This algorithm is extremely useful for our study of random forests in
infinite graphs because in a certain sense, it commutes with passage to the limit. Wilson’s
algorithm can choose a spanning tree at random not only according to uniform measure,
but, in general, proportional to its weight, where, for a spanning tree T , we define its
weight to be
weight(T ) :=
Y
C(e) .
e∈T
To describe Wilson’s method, we define the loop erasure of a path. If P is any finite
path hv0 , v1 , . . . , vl i in G, we define the loop erasure of P, denoted LE(P), by erasing
cycles in P in the order they appear. A slightly different, and more precise, inductive
description of the loop erasure LE(P) = hu0 , u1 , . . . , um i of a path P = hv0 , v1 , . . . , vl i is
as follows. The first vertex u0 of LE(P) is the first vertex v0 of P. Suppose that uj has
been set. Let k be the last index such that vk = uj . Set uj+1 := vk+1 if k < l; otherwise,
let LE(P) := (u1 , . . . , uj ). For future use, note that LE(P) is still well defined when P is
an infinite path that visits no vertex infinitely often.
§3. Wilson’s Method
8
In order to generate a random spanning tree, first pick any vertex r to be the “root”
of the tree. Then create a growing sequence of trees T (i) (i > 0) as follows. Choose any
ordering hv1 , . . . , vn i of the vertices V. Let T (0) := {r}. Suppose that the tree T (i) has
been generated. Start an independent network random walk at vi+1 and stop at the first
time it hits T (i). (If vi+1 ∈ T (i), then the random walk will consist only of hvi+1 i.) Now
create T (i + 1) by adding to T (i) the loop erasure of this random walk. Then T (i + 1) is
a tree. The output of Wilson’s algorithm is the set of edges of the tree T = T (n). (The
root is forgotten.)
Theorem 3.1. (Wilson (1996)) Let (G, C) be a finite network. Wilson’s method yields
a random spanning tree with distribution proportional to weight.
Remark 3.2. In fact, the proof of Theorem 3.1 gives more. It allows the next choice of
vertex vi+1 from which to start the Markov chain to depend on the history of the algorithm
up to that point. This dependence does not affect the distribution of the outcome, provided,
of course, that eventually all vertices are visited.
Wilson’s method for generating spanning trees will also yield (a.s.) a random spanning
tree TG on any recurrent connected network, G. In Proposition 5.6, we shall identify it as
a limit of weighted random spanning trees from finite subnetworks. In particular, when
simple random walk on a graph G is recurrent, we may regard TG as a “uniform random
spanning tree” in G.
Remark 3.3. (WSF on Markov Chains) Wilson’s algorithm is valid in a more general
setup. Let X be a finite irreducible Markov chain. Let V be the set of states of X
and o ∈ V. Define a network structure G on V by letting the directed edges of G be
the pairs [v, u] where there is positive probability to go in one step from v to u; let this
probability be the weight of the directed edge [v, u]. A spanning arborescence T with
root o is a spanning tree of G where each edge belongs to a directed path ending at o.
The weight of T is defined to be the product of the weights of the directed edges in T .
In this setting, Wilson’s algorithm with root o outputs a spanning arborescence T with
distribution proportional to its weight.
Remark 3.4. (Caveat) An automorphism of a Markov chain G is a bijection ϕ from the
state space V of G to itself such that for every v, u ∈ V, the transition probability from v to
u is the same as the transition probability from ϕ(v) to ϕ(u). A Markov chain is transitive
if for each v, u ∈ V, there is an automorphism taking v to u. There are situations where the
reversible Markov chain that arises from a network (G, C) is transitive, but the network
itself is not. For example, let G := Z and set C([n, n + 1]) := 2−n .
§4. Electrical Networks and Random Spanning Trees
9
§4. Electrical Networks and Random Spanning Trees.
In this section, we describe the known connections between random spanning trees and
finite electrical networks, using the approach that will be most fruitful for the extensions to
infinite networks. Throughout this section, (G, C) will denote a finite network. Recall that
P
Cv := {C(e) : e = v} is the conductance of a vertex v in the network and R(e) := 1/C(e)
is the resistance of an edge e.
Let ℓ2 (V) be the real Hilbert space of functions on V with inner product
(f, g)C :=
X
Cv f (v)g(v)
v∈V
and norm kf kC . Since we shall be interested in flows on E, define ℓ2− (E) to be the space
of antisymmetric functions θ on E (i.e., θ(ě) = −θ(e) for each edge e) with inner product
(θ, θ ′ )R :=
X
1X
R(e)θ(e)θ ′ (e) ,
R(e)θ(e)θ ′ (e) =
2
e∈E
e∈E1/2
where E1/2 ⊂ E is a set of oriented edges containing exactly one of each pair e, ě. Given
θ ∈ ℓ2− (E), its energy is E(θ) := (θ, θ)R = kθk2R .
Define the gradient operator ∇ : ℓ2 (V) → ℓ2− (E) by
Note that k∇f kR 6
√
¡
(∇F )(e) := C(e) F (e) − F (e)) .
2kf kC . Define the divergence operator div : ℓ2− (E) → ℓ2 (V) by
(div θ)(v) := Cv−1
X
θ(e) ;
e=v
again, k div θkC 6
√
2kθkR . It is easy to check that −∇ and div are adjoints of each other:
∀F ∈ ℓ2 (V) ∀θ ∈ ℓ2− (E)
(θ, −∇F )R = (div θ, F )C .
A function F : V → R is harmonic at a vertex v if div ∇F (v) = 0, or equivalently (when
P
there are no multiple edges) if Cv F (v) = w∈V C(v, w)F (w).
Given a directed edge e, let χe := 1e − 1ě denote the unit flow along e. Let
⋆ := ∇ ℓ2 (V) ,
P
χe = −∇ 1v . If
that is, ⋆ is the subspace in ℓ2− (E) spanned by the stars
e=v C(e)
Pn
e1 , e2 , . . . , en is an oriented cycle in G, then i=1 χei will be called a cycle. Let ♦ ⊂
§4. Electrical Networks and Random Spanning Trees
10
ℓ2− (E) denote the subspace spanned by these cycles. The subspaces ⋆ and ♦ are clearly
orthogonal to each other (with respect to (•, •)R ). Moreover, the sum of ⋆ and ♦ is all
of ℓ2− (E): Suppose that θ is orthogonal to ♦. Fix a vertex o; for any vertex v ∈ V, define
P
the potential F (v) to be j R(ej )θ(ej ), where e1 , . . . , en is a path from o to v. Since θ
is orthogonal to the cycles, this definition is independent of the choice of path. It follows
that θ = ∇F , as desired.
Given any subspace Z ⊆ ℓ2− (E), let PZ denote the orthogonal projection of ℓ2− (E) onto
Z, and let PZ⊥ denote the orthogonal projection onto the orthogonal complement of Z. Set
I e := P⋆ χe .
Note that div θ = 0 iff θ is orthogonal to ⋆; since I e − χe ⊥ ⋆, it follows that div I e =
div χe = Ce−1 1e − Ce−1 1e . The current I e has least energy among θ ∈ ℓ2− (E) satisfying
div θ = div χe (this is known as “Thomson’s principle”).
Recall that Pv denotes the probability measure of a network random walk starting at
v. The first fundamental relation between random spanning trees, electricity and random
walks is the following, in which the equality of the first and third quantities of (4.1) is due
to Kirchhoff (1847). See Thomassen (1990) for a short combinatorial proof of this equality;
the equality of the second and third quantities is due to Doyle and Snell (1984).
Theorem 4.1. Let T be a random spanning tree of a finite connected network (G, C) and
e, f be edges of G. Let β(e, f ) be the probability that the path in T joining e to e passes
through f in the same direction as f . Consider the network random walk that starts at e
and halts when it hits e. Let J e (f ) be the expected number of times that this walk uses f
minus the expected number of times that it uses fˇ. Then
β(e, f ) − β(e, fˇ) = J e (f ) = I e (f ) .
(4.1)
In particular,
P[e ∈ T ] = Pe [first hit e via traveling along e] = I e (e) .
Proof. Consider loop-erased random walk from e to e. By Wilson’s algorithm, β(e, f ) −
β(e, fˇ) is the expected number of times that loop-erased walk uses f minus the expected
number of times that it uses fˇ. Since every cycle is traversed in each direction an equal
number of times in expectation, this also equals J e (f ). This gives the first equality of
(4.1).
To prove the second equality of (4.1), let F (v) be the expected number of visits of
the network random walk to v. Note that div J e = Ce−1 1e − Ce−1 1e = div I e , which is
§4. Electrical Networks and Random Spanning Trees
11
the same as J e − I e ⊥ ⋆. For any vertex v and any directed edge f , let θv (f ) be the
probability that the first step of the random walk starting at v will use the edge f minus
the probability that it will use the edge fˇ. Thus, Cv θv = −∇1v , whence θv ∈ ⋆. Since
P
J e = v F (v)θv , it follows that J e ∈ ⋆. Since I e ∈ ⋆, it follows that J e = I e , and the
proof is complete.
The matrix of P⋆ in the orthogonal basis {χe : e ∈ E1/2 } is given by
(P⋆ χe , χf )R = (I e , χf )R = R(f )I e (f ) .
(4.2)
In other words, the matrix coefficient at (e, f ) is the voltage difference across f when a
unit current is imposed between the endpoints of e. This matrix is called the transfer
impedance matrix. The related matrix with entries Y (e, f ) := I e (f ) is called the transfer current matrix. Since P⋆ is self-adjoint, the transfer impedance matrix is symmetric.
Therefore Y (e, f )R(f ) = Y (f, e)R(e) (this is called the “reciprocity law”).
Let F be a set of edges. The contracted network G/F is defined by identifying
every pair of vertices that are joined by edges in F . The network G/F may have loops
and multiple edges. We identify the set of edges in G and in G/F . When we need to
indicate the graph G of which T is a subtree, we shall write TG . The contraction operation
is important because of the following easy and well-known observation:
Proposition 4.2. (Contracting Edges) Let G be a finite connected network. Assuming that there is no cycle of G in F , the distribution of TG conditioned on F ⊂ TG is equal
to the distribution of TG/F ∪ F when we think of TG and TG/F as sets of edges.
Next, we examine the effect of contracting edges in the setting of the inner-product
b
b denote the subspace of ℓ2 (E) spanned by the stars of G/F , and let ♦
space ℓ2− (E). Let ⋆
−
b = ♦ + hχF i,
denote the space of cycles (including loops) of G/F . It is easy to see that ♦
b ⊃ ♦ and ⋆
b ⊂ ⋆. Let
where hχF i is the linear span of {χf : f ∈ F }. Consequently, ♦
b ⊂ ⋆ and ⋆
b is the
Z := P⋆ hχF i, which is the linear span of {I f : f ∈ F }. Since ⋆
b we have P⋆ ♦
b = ⋆ ∩ ♦.
b Consequently,
orthogonal complement of ♦,
b = P⋆ ♦
b = P⋆ ♦ + P⋆ hχF i = Z ,
⋆∩♦
and we obtain the orthogonal decomposition
b = ♦ ⊕ Z.
b ⊕ Z and ♦
where ⋆ = ⋆
b ⊕ Z ⊕ ♦,
ℓ2− (E) = ⋆
§4. Electrical Networks and Random Spanning Trees
12
χe
Let e be an edge that does not form a cycle together with edges in F . Set Ibe := P⋆
b ;
this is the analogue of I e in the network G/F . The above decomposition tells us that
⊥
⊥ e
χe
χe
Ibe = P⋆
b = P Z P⋆ = P Z I .
(4.3)
Kirchhoff’s (1847) theorem has the following beautiful generalization due to Burton
and Pemantle (1993):
The Transfer Current Theorem.
Let G be a finite connected network. For any
distinct edges e1 , . . . , ek ∈ G,
P[e1 , . . . , ek ∈ T ] = det[Y (ei , ej )]16i,j6k .
(4.4)
Note that e1 , . . . , ek are unoriented on the left-hand side and are distinct as unoriented
edges. However, an orientation must be chosen for each ei to compute the right-hand side.
Note that the determinant can also be written
P[e1 , . . . , ek ∈ T ] = det[(P⋆ χ̂ei , χ̂ej )R ]16i,j6k ,
where, for each e ∈ E, we define the unit vector χ̂e :=
(4.5)
p
C(e)χe .
The Transfer Current Theorem was shown for the case of two edges in Brooks, Smith,
Stone, and Tutte (1940). The proof here is new.
Proof. If some cycle can be formed from the edges e1 , . . . , ek , then a linear combination of
P
the corresponding columns of [Y (ei , ej )] is zero: suppose that such a cycle is j aj χej ∈ ♦,
where aj ∈ {−1, 0, 1}. Then
X
j
aj R(ej )Y (ei , ej ) =
X
j
´
³
X
ej
ei
ej
χ
χ
= 0,
aj
)R = I ,
aj (I ,
ei
j
R
because I ei ⊥ ♦. Therefore, both sides of (4.4) are 0. For the remainder of the proof, we
may assume that there are no such cycles.
Since P⋆ is self-adjoint and its own square, (4.2) gives that for any two edges e and
f,
Y (e, f ) = C(f )(P⋆ χe , χf )R = C(f )(P⋆ χe , P⋆ χf )R = C(f )(I e , I f )R .
Therefore,
det[Y (ei , ej )]16i,j6k =
Ã
k
Y
i=1
!
C(ei ) det Yk ,
(4.6)
§4. Electrical Networks and Random Spanning Trees
13
where Yk is the Gram matrix with entries (I ei , I ej )R . The determinant of a Gram matrix
is the squared volume of the parallelepiped spanned by its determining vectors, whence
det[Y (ei , ej )]16i,j6k =
k
Y
i=1
°2
°
C(ei ) °PZ⊥i I ei °R ,
where Zi is the linear span of I e1 , . . . , I ei−1 .
From Proposition 4.2, we know that P[ei ∈ T | e1 , . . . , ei−1 ∈ T ] = Ibei (ei ) in the
graph G/{e1 , . . . , ei−1 }. Applying (4.6) and (4.3) gives that
Therefore,
°2
°
Ibei (ei ) = C(ei )(Ibei , Ibei )R = C(ei ) °PZ⊥i I ei °R .
P[e1 , . . . , ek ∈ T ] =
=
k
Y
i=1
k
Y
i=1
P[ei ∈ T | e1 , . . . , ei−1 ∈ T ]
°2
°
C(ei ) °PZ⊥i I ei °R = det[Y (ei , ej )]16i,j6k .
An extension of the Transfer Current Theorem is as follows. For a set of unoriented
edges B and a linear map P , write
P
B,e
:=
½
P
id − P
if e ∈ B,
if e ∈
/ B,
(4.7)
where id is the identity map. As shown in Cor. 4.4 of Burton and Pemantle (1993), if G
is a finite network and B ⊆ K are sets of unoriented edges, then
´ i
h³
B,e χ̂e χ̂e′
,
P[T ∩ K = B] = det P⋆
.
(4.8)
R e,e′ ∈K
(Again, to compute the right-hand side, an orientation must be chosen for each edge in
K.) Indeed, the identity
h³
´ i
′
det (P⋆ + xe id)χ̂e , χ̂e
R e,e′ ∈K
=E
hY
e∈K
(1{e∈T } + xe )
i
(4.9)
is easily verified by comparing coefficients of each monomial in the variables hx e i: the
Q
coefficient of e∈S xe on the left-hand side of (4.9) equals P[K \ S ⊂ T ] by (4.5). Applying
(4.9) with xe = 0 if e ∈ B and xe = −1 if e ∈
/ B, then multiplying by (−1)|K\B| , we obtain
(4.8).
We shall need the following special case of Rayleigh’s monotonicity principle:
§4. Electrical Networks and Random Spanning Trees
Rayleigh’s Monotonicity Principle.
14
Let (G, C) be a finite network and let e be an
e
the current I e in the network H.
edge in G. Denote by IH
e
e
(a) If G′ is a subgraph of G that contains e, then IG
′ (e) > IG (e).
e
e
(b) If F ⊂ E is such that F ∪ {e} has no cycles containing e, then I G/F
(e) 6 IG
(e).
Proof. Appending edges to a network or contracting edges in it can only increase the subspace ♦, hence can only decrease the norm of P♦⊥ χe = I e . Since I e (e) = C(e)(I e , I e )R =
C(e)E(I e ) by (4.6), Rayleigh’s principle follows.
Corollary 4.3. Let (G, C) be a finite connected network and let F ⊂ E.
(a) If G′ is a subgraph of G that contains F , then
P[F ⊂ TG′ ] > P[F ⊂ TG ] .
(b) For any two distinct edges e and f , we have P[f ∈ TG | e ∈ TG ] 6 P[f ∈ TG ]. More
generally, if F ′ ⊂ E is disjoint from F , then P[F ⊂ TG/F ′ ] 6 P[F ⊂ TG ] .
Proof. The corollary follows from Rayleigh’s principle using Theorem 4.1, Proposition 4.2
and induction on |F |.
An event A ⊆ 2E is called increasing if F1 ⊂ F2 ⊆ E and F1 ∈ A imply F2 ∈ A. We
say that A ignores a set F ⊆ E if F1 \ F = F2 \ F and F1 ∈ A imply F2 ∈ A. Feder and
Mihail (1992) proved:
Theorem 4.4. (Negative Correlations)
Let e ∈ E and suppose that A ⊆ 2 E is
increasing and ignores {e}. Then P[T ∈ A | e ∈ T ] 6 P[T ∈ A].
For the convenience of the reader, we reproduce the proof.
Proof. We induct on the sum |V| + |E| for G. The case |V| = 2 is trivial, but it is also
the only place we explicitly use the assumption that A is increasing. Now assume that
|V| > 3 and that we know the result for graphs where the sum of the number of vertices
and the number of edges is smaller than in G. Fix an edge e of G. Since e becomes a loop
in the contraction G/e, every spanning tree of G/e has |V| − 2 edges and does not contain
e. Thus, given A and e, we have
X
f ∈E\e
P[A, f ∈ T | e ∈ T ] = (|V| − 2)P[A | e ∈ T ] = P[A | e ∈ T ]
X
f ∈E\e
P[f ∈ T | e ∈ T ] .
Therefore, there is some f ∈ E \ e such that P[A | f, e ∈ T ] > P[A | e ∈ T ]. This also
means that
P[A | f, e ∈ T ] > P[A | f ∈
/ T, e ∈ T ] .
(4.10)
§4. Electrical Networks and Random Spanning Trees
15
Now
P[A | e ∈ T ] = P[f ∈ T | e ∈ T ]P[A | f, e ∈ T ] + P[f ∈
/ T | e ∈ T ]P[A | f ∈
/ T, e ∈ T ] .
Corollary 4.3(b) implies that
P[f ∈ T | e ∈ T ] 6 P[f ∈ T ] .
(4.11)
The event A/f := {H ⊆ E : H ∪ {f } ∈ A} on the network G/f is increasing and ignores
{e}, whence applying the induction hypothesis to it yields
P[A | f, e ∈ T ] 6 P[A | f ∈ T ] .
(4.12)
Similarly, the induction hypothesis applied to the event A\f := {H ⊆ E \ f : H ∈ A} on
the network G \ f gives
P[A | f ∈
/ T, e ∈ T ] 6 P[A | f ∈
/ T].
(4.13)
From (4.11) and (4.10), we have
P[A | e ∈ T ] 6 P[f ∈ T ]P[A | f, e ∈ T ] + P[f ∈
/ T ]P[A | f ∈
/ T, e ∈ T ] ;
(4.14)
we have replaced a convex combination in (4.10) by another in (4.14) that puts more weight
on the larger term. By (4.12) and (4.13), we have that the right-hand side of (4.14) is
6 P[f ∈ T ]P[A | f ∈ T ] + P[f ∈
/ T ]P[A | f ∈
/ T ] = P[A] .
Remark 4.5. More generally, if A and B are both increasing and they depend on disjoint
sets of edges (i.e., there is a set of edges F such that A ignores F and B ignores the
complement of F ), then the events {T ∈ A} and {T ∈ B} are negatively correlated. See
Feder and Mihail (1992).
Conjecture 4.6. (BK-Type Inequality) We say that A, B ⊂ 2 E occur disjointly
for F ⊂ E if there are disjoint sets F1 , F2 ⊂ E such that F ′ ∈ A for every F ′ with
F ′ ∩ F1 = F ∩ F1 and F ′ ∈ B for every F ′ with F ′ ∩ F2 = F ∩ F2 . Let A, B ⊂ 2E be
increasing. Then the probability that A and B occur disjointly for the random spanning
tree T is at most P[T ∈ A]P[T ∈ B]. (The BK inequality of van den Berg and Kesten
(1985) says that the same is true when T is a random subset of E chosen according to any
product measure on 2E .)
§5. Basic Properties of Random Spanning Forests
16
§5. Basic Properties of Random Spanning Forests.
Let (G, C) be an infinite connected network, and let V be the vertices of G. Let
S∞
V1 ⊂ V2 ⊂ · · · be finite connected subsets of V with n=1 Vn = V. Let Gn = (Vn , En )
be the subgraph spanned by Vn ; that is, an edge of G appears in En if its endpoints
are in Vn . Then hGn i is called an exhaustion of G. Let µF
n be the weighted spanning
tree probability measure on Gn (the superscript F stands for “free” and will be explained
below). Given a finite set B of edges, we have B ⊆ En for large enough n. For such n, we
have by Corollary 4.3 that
F
µF
n (B ⊆ T ) > µn+1 (B ⊆ T ) .
In particular, the limit µF (B ⊆ T ) := limn→∞ µF
n (B ⊆ T ) exists. It follows from the
inclusion-exclusion principle that for any finite B ⊆ K ⊂ E, the limit µ F (T ∩ K = B) :=
F
limn→∞ µF
n (T ∩K = B) exists. Thus, µ is defined on all elementary cylinders. This allows
us to define µF on cylinder events, i.e., finite (disjoint) unions of elementary cylinders, and
hence uniquely defines a probability measure µF on 2E . We call µF the (weighted) free
spanning forest measure on G and denote it FSF, since clearly it is carried by the set of
spanning forests of G. In the case where all the edges of G have equal weight, we call µ F
the free uniform spanning forest.
It is easily seen that µF does not depend on the exhaustion {Gn }. Indeed, let {G′n }
be another such exhaustion, and construct inductively an exhaustion {G ′′n } that contains
infinitely many graphs from {Gn } and from {G′n }. Since the limit measure µF exists for
the exhaustion {G′′n }, it follows that the limit is the same for {G′n } as for {Gn }.
There is another natural way of taking limits of spanning trees. In disregarding
the complement of Gn , we are (temporarily) disregarding the possibility that a spanning
tree or forest of G may connect the boundary vertices of Gn outside of Gn in ways that
would affect the possible connections within Gn itself. An alternative approach forces all
connections outside of Gn : Let GW
n be the graph obtained from G by contracting the
vertices outside Gn to a single vertex, zn . (In GW
n , the conductance Czn of zn may be
infinite. However, the sum of the conductances of the edges incident with z n that are not
loops is finite, and therefore the infinite conductance of zn does not cause any problems.)
W
W
W
Let µW
n be the random spanning tree measure on Gn . Since Gn is obtained from Gn+1
by contracting edges, µW
n (B ⊆ T ) is increasing in n by Corollary 4.3. Thus, we may again
define the limiting probability measure µW , which does not depend on the exhaustion. It
is called the (weighted) wired spanning forest and denoted WSF. When all the edges
of G have equal weight, we call µW the wired uniform spanning forest. The term
§5. Basic Properties of Random Spanning Forests
17
“wired” comes from thinking of GW
n as having its boundary wired together. In statistical
mechanics, measures on infinite configurations are also defined by taking limits from finite
graphs with appropriate boundary conditions. The terms “free” and “wired” originate
there. If G is itself a tree, the free spanning forest is obviously concentrated on just {G},
while the wired spanning forest is usually more interesting (see Remark 5.7). When the
free and the wired uniform spanning forests agree, we sometimes drop the terms “free”
and “wired”.
As we shall see, the WSF is much better understood than the FSF. Indeed, there
is a direct construction of it that avoids weak limits: Let (G, C) be a transient network.
Define F0 = ∅. Inductively, for each n = 1, 2, . . ., pick a vertex vn and run a network
random walk starting at vn . Stop the walk when it hits Fn−1 , if it does, but otherwise
let it run indefinitely. Let Pn denote this walk. Since G is transient, with probability 1,
Pn visits no vertex infinitely often, so LE(Pn ) is well defined. Set Fn := Fn−1 ∪ LE(Pn )
S
and F := n Fn . Assume that the choices of the vertices vn are made in such a way that
{v1 , v2 , . . .} = V. The same reasoning as in Wilson’s proof of Theorem 3.1 shows that the
resulting distribution of F is independent of the order in which we choose starting vertices.
We shall refer to this method of generating a random spanning forest as Wilson’s method
rooted at infinity.
Theorem 5.1. (WSF through Wilson’s Method) The wired spanning forest on any
transient network G is the same as the random spanning forest generated by Wilson’s
method rooted at infinity.
¡
¢
Proof. For any path hxk i that visits no vertex infinitely often, LE hxk : k 6 Ki →
¡
¢
¡
¢
LE hxk : k > 0i as K → ∞. That is, if LE hxk : k 6 Ki = huK
i : i 6 mK i and
¡
¢
LE hxk : k > 0i = hui : i > 0i, then for each i and all large K, we have uK
i = ui ; this
¡
follows from the definition of loop erasure. Since G is transient, it follows that LE hX(k) :
¢
¡
¢
k 6 Ki → LE hX(k) : k > 0i as K → ∞ a.s., where hX(k)i is a random walk starting
from any fixed vertex.
Let Gn be an exhaustion of G and GW
n the graph formed by contracting the vertices
outside Gn to a vertex zn . Let T (n) be a random spanning tree on GW
n and F the limit
of T (n) in law. Given e1 , . . . , eM ∈ E, let hXvi (k)i be independent random walks starting
from the endpoints v1 , . . . , vL of e1 , . . . , eM . Run Wilson’s algorithm rooted at zn from
the vertices v1 , . . . , vL in that order; let τjn be the time that hXvj (k)i reaches the portion
§5. Basic Properties of Random Spanning Forests
18
of the spanning tree created by the preceding random walks hXvl (k)i (l < j). Then
L
[
¡
¢
LE hXvj (k) : k 6 τjn i for 1 6 i 6 M .
P[ei ∈ T (n) for 1 6 i 6 M ] = Pei ∈
j=1
Let τj be the stopping times corresponding to Wilson’s method rooted at infinity. By
induction on j, we see that τjn → τj as n → ∞, so that
L
[
¡
¢
LE hXvj (k) : k 6 τj i for 1 6 i 6 M .
P[ei ∈ F for 1 6 i 6 M ] = Pei ∈
j=1
That is, F has the same law as the random spanning forest generated by Wilson’s method
rooted at infinity.
Definition 5.2. (Oriented WSF) Let G be a transient network and use Wilson’s method
rooted at ∞ to get the wired spanning forest F. For every edge e of F, choose the orientation
that agrees with the direction of the loop-erased walk of the method that inserted e into
F. Call the resulting oriented graph the wired spanning forest oriented towards
infinity, and let OWSF denote its law.
Proposition 5.3. (Automorphism Invariance) FSF and WSF are invariant under any
automorphisms that the network may have. If the network is transient, then the OWSF is
also automorphism invariant.
Proof. The claim regarding FSF and WSF is clear, since we have shown that they do not
depend on the exhaustion. To establish the invariance of the OWSF, one needs to show
only that when using Wilson’s method rooted at infinity, the order in which the starting
vertices of the random walks are picked does not affect the distribution of the forest. Since
this holds for finite graphs, the invariance of the OWSF follows by taking an exhaustion of
G and using the proof of Theorem 5.1. Alternatively, the proof of Wilson’s algorithm in
Propp and Wilson (1998) also applies to the OWSF on a transient network.
Remark 5.4. Wilson’s method rooted at infinity can be performed on any transient
Markov chain, and the law of the resulting forest does not depend on the order in which
the vertices are chosen; this follows from the proof of Theorem 3.1 as given in Propp and
Wilson (1998). We use WSF to denote this forest on G.
Proposition 5.5. Let G be a locally finite infinite connected network. For both FSF and
WSF, all component trees are infinite a.s.
Proof. For any specific finite subtree t in G, the event that all the edges incident to t are
W
absent is assigned probability 0 by µF
n and µn , provided n is sufficiently large. Since there
are only countably many such events, this establishes the proposition.
§5. Basic Properties of Random Spanning Forests
19
Proposition 5.6. (Equality in Recurrent Networks) If G is an infinite recurrent
network, then the random spanning tree TG generated by using Wilson’s method on G (with
any choice of root r and any ordering of the vertices) coincides in distribution with the
WSF and the FSF. In particular, the distribution of TG does not depend on the choice of
root nor on the ordering of the vertices.
Proof. Consider an exhaustion hGn i of G by finite networks.
We must show
¯
¯ that for any
¯
¯
event B ∈ 2E depending on only finitely many edges, ¯P[TG ∈ B] − µW
n [B]¯ → 0 as n → ∞
(and similarly for µF
n ). Let K0 be the set of vertices incident to the edges on which B
depends. Let K be the union of K0 and the set of vertices that precede some vertex in K0
in the ordering given in the hypothesis. Denote by ∂V Gn the vertex boundary of Gn ,
i.e., the set of vertices not in Gn that are adjacent to some vertex in Gn . By examining
Wilson’s algorithm, we see that
¯
¯ X
¯
¯
W
Pv [τ∂V Gn < τr ] ,
¯P[TG ∈ B] − µn [B]¯ 6
v∈K
and the right-hand side tends to 0 as n → ∞ by recurrence. This argument also applies
to µF
n.
Remark 5.7. It is easy to see that on any transient tree with no transient ray, there is
an edge such that removing it breaks the tree into two transient components. Thus by
Wilson’s method, when G is a tree with no transient ray, the FSF coincides with the WSF
iff G is recurrent. This was first proved by Häggström (1998).
The FSF and WSF also coincide in many transient networks (e.g., in Z d for d > 3); in
Theorem 7.3, we shall determine precisely when this happens. In all cases, though, there
is a simple inequality between these two probability measures:
∀e ∈ E FSF(e ∈ F) > WSF(e ∈ F)
(5.1)
W
since µF
n (e ∈ T ) > µn (e ∈ T ) by Corollary 4.3. More generally, by repeated use of
Theorem 4.4, for every increasing A ⊆ 2E that ignores all but finitely many edges, we have
FSF(F ∈ A) > WSF(F ∈ A) .
We therefore say that FSF stochastically dominates WSF. By Strassen’s (1965) theorem, this inequality implies that there is a monotone coupling of the two measures, FSF
and WSF, in the sense that there is a probability measure on the set
{(F1 , F2 ) : Fi is a spanning forest of G and F1 ⊆ F2 }
that projects in the first coordinate to WSF and in the second to FSF.
§6. Average and Expected Degrees
20
Remark 5.8. Because of the monotone coupling, the number of trees in the FSF on a
network is stochastically dominated by the number in the WSF. If these two numbers are
a.s. finite and equal, then FSF = WSF.
Remark 5.9. Similarly, if each component of the FSF has a.s. one end, then FSF = WSF,
because a lower bound for the number of ends of an FSF-component is the number of WSF
components that it contains in a monotone coupling that gives FSF ⊇ WSF.
Proposition 5.10. If E[degF (v)] is the same under FSF and WSF for every v ∈ V, then
FSF = WSF.
Proof. In the monotone coupling described above, the set of edges adjacent to a vertex v
in the WSF is a subset of those adjacent to v in the FSF. The hypothesis implies that for
each v, these two sets coincide a.s.
Remark 5.11. It follows that if FSF and WSF agree on single-edge probabilities, i.e., if
equality holds in (5.1) for all e ∈ E, then FSF = WSF. This is due to Häggström (1995).
§6. Average and Expected Degrees.
Let G be a graph. For V′ ⊂ V, let
∂V′ := {e ∈ E : e ∈ V′ , e ∈
/ V′ } .
We say that G = (V, E) is amenable if there is an exhaustion V1 ⊂ V2 · · · ⊂ Vn ⊂ · · · ⊂ V
with
lim |∂Vn |/|Vn | = 0 .
n→∞
Thus, a finitely generated group is amenable iff its Cayley graph is. Every finitely generated
abelian group is amenable. A network is called amenable if its underlying graph is.
Remark 6.1. (Average Degrees in Amenable Networks) (Compare Theorem 3.2 in
Thomassen (1990).) Let G be an amenable infinite network as witnessed by the exhaustion
hVn i. Let F be any deterministic spanning forest of G all of whose components (trees) are
infinite. Then the average degree of vertices in F is 2. More precisely, if deg F (v) denotes
the degree of v in F, then
lim |Vn |−1
n→∞
X
v∈Vn
degF (v) = 2 ,
§6. Average and Expected Degrees
21
and the limit is uniform in F. This is because the number of components of F intersecting
Vn is at most |∂Vn | and a tree with k vertices has k − 1 edges.
Remark 6.2. Let G be an amenable transitive connected infinite graph. Let F be the free
or the wired uniform spanning forest on G. Then by Remark 6.1 and Proposition 5.5, for
every v ∈ V, the expected degree of v in F is 2.
The following is essentially due to Häggström (1995).
Corollary 6.3. On any transitive amenable network, FSF = WSF.
Proof. By transitivity and Remark 6.2, E[deg F (v)] = 2 for both FSF and WSF. Apply
Proposition 5.10.
Although the transitivity assumption cannot be dropped (see, e.g., Example 9.3), the
amenability assumption is not needed to determine the expected degree in the WSF on a
transitive network:
Theorem 6.4. In a transitive network G, the WSF-expected degree of every vertex is 2.
Proof. If G is recurrent, then it is amenable (Dodziuk 1984), and the result follows from
Remark 6.2. So assume that G is transient. In the oriented wired spanning forest OWSF,
the out-degree of every vertex is 1. We need to show that the expected in-degree of every
vertex is 1. For this, it suffices to prove that
OWSF[f ∈ F] = OWSF[fˇ ∈ F]
(6.1)
for every directed edge f . Set
α(e) := C(e)Pe [τe = ∞] .
Let f be a directed edge. Start a network random walk hXv (n)i at v := f . For each edge
e satisfying e = v, the probability that the first step of the walk is e and the walk does not
return to v is α(e)/Cv . Therefore, OWSF[f ∈ F], which is the probability that f will be in
LEhXv (n)i, is given by
α(f )
P
.
e=v α(e)
The denominator does not depend on v by transitivity, so to prove (6.1) it suffices to verify
P
that α(f ) = α(fˇ). By reversibility, the Green function g(v, u) :=
Pv [Xv (n) = u]
n>0
satisfies
Cv Pv [τu < ∞]g(u, u) = Cv g(v, u) = Cu g(u, v) = Cu Pu [τv < ∞]g(v, v)
§7. Potential Theory
22
for any u, v ∈ V, whence transitivity implies
Pv [τu < ∞] = Pu [τv < ∞] .
(6.2)
Thus α(f ) = α(fˇ) and (6.1) follows.
Recall from Remark 5.4 that the WSF can be constructed using Wilson’s method
rooted at ∞ on any transient Markov chain; however, the conclusion of Theorem 6.4 does
not always hold for transitive transient Markov chains that are not reversible: Consider an
irreducible chain on a 3-regular tree T (3) with a distinguished end ξ, where the transition
probability from a vertex v to its “parent” (the unique neighbor closer to ξ) is greater than
1/2; the resulting WSF contains all edges of T (3) . For another example, if the transition
probability to each “child” is 1/2 − ǫ, then the expected degree of any vertex is 3/2 + O(ǫ).
Nevertheless, as we show next, the reversibility assumption in Theorem 6.4 can be
replaced by unimodularity of the automorphism group of G. Recall that a locally com-
pact group is called unimodular if its left Haar measure is also right invariant. See
BLPS (1999) for more details on unimodular automorphism groups and its significance for
random subgraphs.
Theorem 6.5. Let G be a transient Markov chain and assume that the automorphism
group of G is transitive and unimodular. Then the WSF-expected degree of every vertex in
G is 2.
Proof. Consider the OWSF. The out-degree of every vertex is 1. To compute the expected
in-degree, we use the Mass-Transport Principle (BLPS (1999)). Transport a unit mass
from a vertex v to the vertex w if there is a directed edge from v to w. By the MassTransport Principle, the expected mass transported to v is the expected mass transported
from v, which is 1. Hence the expected in-degree is 1.
§7. Potential Theory.
We turn now to an electrical criterion for the equality of FSF and WSF and develop
the associated potential theory. There are two natural ways of defining currents between
vertices of an infinite graph, corresponding to the two ways of defining spanning forests. We
shall define these currents using Hilbert space projections and recall how they correspond
to limits of currents on finite subgraphs.
Let G be an infinite network. As in Section 4, for antisymmetric functions θ, θ ′ : E →
R, set
(θ, θ ′ )R :=
X
1X
R(e)θ(e)θ ′ (e) ,
R(e)θ(e)θ ′ (e) =
2
e∈E
e∈E1/2
§7. Potential Theory
23
and let ℓ2− (E) be the Hilbert space of all antisymmetric functions θ with E(θ) = (θ, θ) R <
∞. Let ⋆ denote the closure in ℓ2− (E) of the linear span of the stars and ♦ the closure
of the linear span of the cycles. Since every star and every cycle are orthogonal, it is still
true that ⋆ ⊥ ♦. However, it is no longer necessarily the case that ℓ2− (E) = ⋆ ⊕ ♦; in
fact, we shall see that this is equivalent to FSF = WSF. Thus, we are led to define two
possibly different currents,
IFe := P♦⊥ χe ,
the free current between the endpoints of e (also called the “limit current”), and
e
IW
:= P⋆ χe ,
the wired current between the endpoints of e (also called the “minimal current”). The
names for these currents are explained by the following two well-known propositions. The
first is proved by noting that the space of cycles of Gn increases to ♦, while the second
follows from the fact that the space of stars of GW
n increases to ⋆. See, e.g., Soardi (1994),
Cor. 3.17 and Thm. 3.25 for more details.
Proposition 7.1. (Free Currents) Let G be an infinite network exhausted by finite
e
. Then kIn − IFe kR → 0 as n → ∞
subnetworks hGn i. Let e be an edge in G1 and In := IG
n
and E(IFe ) = IFe (e)R(e).
Proposition 7.2. (Wired Currents) Let G be an infinite network exhausted by finite
subnetworks hGn i. Let GW
n be formed by identifying the complement of Gn to a single
e
e
vertex. Let e be an edge in G1 and In := IG
Then kIn − IW
kR → 0 as n → ∞
W.
n
e
e
) = IW
and E(IW
(e)R(e), which is the minimal energy among all θ ∈ ℓ2− (E) satisfying
div θ = div χe .
e
e
e
Since E(IW
) 6 E(IFe ) with equality iff IW
= IFe , we obtain that IW
(e) 6 IFe (e) with
e
equality iff IW
= IFe .
Recall that a function F on V is harmonic if div ∇F = 0. A function F is a Dirichlet
function if it has finite Dirichlet energy E(∇F ). The collection of harmonic Dirichlet
functions on V is denoted HD(G), or simply HD.
Suppose that θ ∈ ℓ2− (E) is orthogonal to ⋆ and to ♦. As we have seen in Section 4,
it follows from θ ∈ ♦⊥ that there is a function F such that θ = ∇F , and hence also F
has finite Dirichlet energy. Since θ ∈ ⋆⊥ , we have div θ = 0, so F ∈ HD. Therefore
⋆⊥ ∩ ♦⊥ ⊆ ∇HD. Conversely, it is immediate that ∇HD is orthogonal to ⋆ ⊕ ♦. This
gives the orthogonal decomposition
ℓ2− (E) = ⋆ ⊕ ♦ ⊕ ∇HD .
(7.1)
§7. Potential Theory
24
On every network, the constant functions are in HD. For some networks G, these
are the only functions in HD(G); in that case, we write HD(G) ∼
= R. For example,
Thomassen (1989) proved that if G is a Cartesian product of two infinite graphs, then
HD(G) ∼
= R; see also Soardi (1994), Thm. 4.17. Cayley graphs of Kazhdan groups G also
satisfy HD(G) ∼
= R; see Bekka and Valette (1997) for this and a summary (Thm. D) of
other groups with this property. See also Remark 7.5 below.
Theorem 7.3. For any network G, the following are equivalent:
(i) FSF = WSF;
e
(ii) IW
= IFe for every edge e;
(iii) ℓ2− (E) = ⋆ ⊕ ♦;
(iv) HD(G) ∼
= R.
Proof. From Theorem 4.1 and Propositions 7.1 and 7.2, we have that FSF(e ∈ T ) = I Fe (e)
e
and WSF(e ∈ T ) = IW
(e). Now use Remark 5.11 to deduce that (i) and (ii) are equivalent.
For the next equivalence, note that ℓ2− (E) = ⋆ ⊕ ♦ is equivalent to P⋆ = P♦⊥ . Since
{χe : e ∈ E1/2 } is a basis for ℓ2− (E), this is also equivalent to P⋆ χe = P♦⊥ χe for all edges
e. That (iii) and (iv) are equivalent follows from (7.1).
Remark 7.4. Doyle (1988) proved that (ii) and (iv) are equivalent.
Remark 7.5. From Corollary 6.3 and Theorem 7.3, we obtain that every transitive amenable network G satisfies HD(G) ∼
= R. This result is due to Medolla and Soardi (1995).
See Benjamini, Lyons and Schramm (1999) for more applications of Theorem 7.3 to the
study of harmonic Dirichlet functions.
Definition 7.6. A rough isometry is a (not necessarily continuous) map
ϕ : (X, distX ) → (Y, distY )
between metric spaces such that for some constant K > 0 and every x, x′ ∈ X,
and
¡
¢
K −1 distX (x, x′ ) − K 6 distY ϕ(x), ϕ(x′ ) 6 K distX (x, x′ ) + K
¡
¢
sup inf distY ϕ(x), y < ∞ .
y∈Y x∈X
When X or Y is a network on a graph G, the metric will be assumed to be the distance
in the graph.
§7. Potential Theory
25
Theorem 7.7. (Soardi (1993)) Let G and G′ be two networks with conductances C and
C ′ . Suppose that C, C ′ , C −1 , C ′−1 are all bounded, that the degrees in G and G′ are all
bounded, and that ϕ : V → V′ is a rough isometry. Then HD(G) ∼
= R.
= R iff HD(G′ ) ∼
Therefore, if G and G′ are roughly isometric networks with bounded edge conductance
and resistance, then the wired and free spanning forests coincide on one iff they do on the
other. However, this does not mean that the basic topologies of the spanning forests must
be the same. Indeed, rough isometries, even merely bounded changes of conductance,
can change the wired spanning forest from a single tree to infinitely many trees: see the
example used to prove Thm. 3.5 in Benjamini and Schramm (1996a). On the other hand,
changing the generators in a Cayley graph cannot have this effect: see Corollary 9.6.
e
Let YF (e, f ) := IFe (f ) and YW (e, f ) := IW
(f ) be the free and wired transfer current
matrices.
Theorem 7.8. (The Transfer Current Theorem for Infinite Graphs) Given
any network G and any distinct edges e1 , . . . , ek ∈ G, we have
FSF[e1 , . . . , ek ∈ T ] = det[YF (ei , ej )]16i,j6k
and
WSF[e1 , . . . , ek ∈ T ] = det[YW (ei , ej )]16i,j6k .
Proof. This is immediate from the Transfer Current Theorem of Section 4 and Propositions
7.1 and 7.2.
By (4.5), another way to write these equations is
FSF[e1 , . . . , ek ∈ T ] = det[P♦⊥ (χ̂ei , χ̂ej )]16i,j6k
and
WSF[e1 , . . . , ek ∈ T ] = det[P⋆ (χ̂ei , χ̂ej )]16i,j6k .
§8. Ergodic Properties
26
§8. Ergodic Properties.
An easy consequence of Theorem 7.8 is mixing, hence ergodicity if the automorphism
group acts on the network with an infinite orbit:
Corollary 8.1. (Mixing) For any infinite network and P = FSF or WSF, let A be a
cylinder event, k > 1, and hBn i be a sequence of cylinder events each depending on at most
k edges. If, for each n, all the edges on which Bn depends are at distance at least n from
the edges on which A depends, then
¯
¯
lim ¯P[A ∩ Bn ] − P[A]P[Bn ]¯ = 0 .
n→∞
Proof. For a finite set of edges K, let y(K) := det[(P χ̂e , χ̂f )]e,f ∈K , where P = P♦⊥ or P⋆ ,
as appropriate. Since (P χ̂e , χ̂f ) → 0 as dist(e, f ) → ∞ for fixed e, it follows that for any
finite K and any k > 1,
¯
¯
lim sup{¯y(K ∪ K ′ ) − y(K)y(K ′ )¯ : |K ′ | 6 k, dist(K, K ′ ) > n} = 0 ,
n→∞
i.e.,
¯
¯
lim sup{¯P[K ⊆ F, K ′ ⊆ F] − P[K ⊆ F]P[K ′ ⊆ F]¯ : |K ′ | 6 k, dist(K, K ′ ) > n} = 0 .
n→∞
Since every cylinder event can be expressed in terms of such elementary cylinder events,
we get the result for all cylinder events.
Corollary 8.2. Let G be an infinite network such that Aut(G) has an infinite orbit.
Then WSF and FSF are ergodic measures for the action of Aut(G). If WSF and FSF on G
are distinct, then they are singular measures on the space 2E .
We do not know if the singularity assertion above holds without the hypothesis on
Aut(G); see Question 15.11.
Proof. The hypothesis implies that every orbit of Aut(G) is infinite. If A is an Aut(G)
invariant event, then approximating A by cylinder sets, we see from Corollary 8.1 that A is
independent of itself, so it has probability 0 or 1. Finally, distinct ergodic measures under
any group action are always singular; see, e.g., Furstenberg (1981).
In fact, we have a much stronger property than mixing, namely, tail triviality. For a
set of edges K ⊆ E, let F(K) denote the σ-field of events depending only on K. Define
the tail σ-field to be the intersection of F(E \ K) over all finite K. We say that a measure
§8. Ergodic Properties
27
on 2E has trivial tail if every event in the tail σ-field has measure either 0 or 1. Recall
that tail triviality is equivalent to
∀A1 ∈ F(E) ∀ǫ > 0 ∃K finite ∀A2 ∈ F(E \ K)
(See, e.g., Georgii (1988), p. 120.)
¯
¯
¯P[A1 ∩ A2 ] − P[A1 ]P[A2 ]¯ < ǫ . (8.1)
Pemantle (1991) proved that if FSF = WSF, then the tail σ-field of the spanning forest
is trivial. R. Solomyak (1999) independently observed Corollary 8.1 and showed that the
tail σ-field of the free spanning forest is trivial on the Cayley graph of any Fuchsian group
(with the standard generators) that is not co-compact. In fact, as we now show, the tails
of both the free and the wired spanning forests on every network are trivial. We have two
proofs of this, one short and qualitative that applies negative correlations, and a longer
proof that yields a quantitative correlation bound that refines Corollary 8.1.
Theorem 8.3. The WSF and FSF have trivial tail on every network.
Proof. Let G be an infinite network exhausted by finite subnetworks hGn i. Recall from
W
Section 5 that µF
n denotes the weighted spanning tree measure on Gn and µn denotes
the weighted spanning tree measure on the “wired” graph GW
n . Let νn be any “partially
wired” measure, i.e., the weighted spanning tree measure on a graph G ∗n obtained from a
finite network G′n satisfying Gn ⊂ G′n ⊂ G by contracting some of the edges in G′n that
are not in Gn . Repeated applications of Theorem 4.4 give that any increasing event B
measurable with respect to edges in Gn satisfies
F
µW
n (B) 6 νn (B) 6 µn (B) .
(8.2)
Let M > n and let A be a cylinder event that is measurable with respect to the edges in
GM \ Gn and such that µW
M (A) > 0. For each event B as in (8.2), we have
W
µW
n (B) 6 µM (B | A) .
(8.3)
To see this, condition separately on each possible configuration of edges of G M \ Gn that
is in A, and use (8.2). Fixing A and letting M → ∞ in (8.3) gives
µW
n (B) 6 WSF(B | A) .
(8.4)
This applies to all cylinder events A that are measurable with respect to the complement
of Gn with WSF(A) > 0, and therefore the assumption that A is a cylinder event can be
dropped. Thus (8.4) holds for all tail events A of positive probability. Taking n → ∞
there gives
WSF(B) 6 WSF(B | A) ,
(8.5)
§8. Ergodic Properties
28
where B is any increasing cylinder event and A is any tail event. Thus, (8.5) also applies
to the complement Ac . Since WSF(B) = WSF(A)WSF(B | A) + WSF(Ac )WSF(B | Ac ), it
follows that WSF(B) = WSF(B | A). Therefore, every tail event A is independent of every
increasing cylinder event, whence A is trivial. The argument for the FSF is similar.
Next, we consider the quantitative version of tail triviality. For any network G and
F ⊆ E, let hχF i denote the closed linear span of {χf : f ∈ F } and set PF := PhχF i ,
PF⊥ := Ph⊥χF i .
Theorem 8.4. Let T be a weighted random spanning tree on a finite network G. Let F
and K be disjoint nonempty sets of edges. Let B be a subset of K. Then
X
¡
¢
Var P[T ∩ K = B | T ∩ F ] 6 |K|
C(e)kPF I e k2R .
(8.6)
e∈K
If A1 ∈ F(K) and A2 ∈ F(F ), then
|P[A1 ∩ A2 ] − P[A1 ]P[A2 ]| 6
Ã
22|K| |K|
X
e∈K
C(e)kPF I e k2R
!1/2
.
(8.7)
Before proving Theorem 8.4, we explain why it implies Theorem 8.3. In fact, we show
the more quantitative (8.1). Let G be an infinite network, let P be WSF or FSF on G, as
e
appropriate, and let I e be IW
or IFe , respectively. Then (8.7) extends to give the same
inequality on G by taking limits over an exhaustion of G. Let A be any event and ǫ > 0.
Find a finite set K1 and A1 ∈ F(K1 ) such that P[A1 △A] < ǫ/2. Now find a finite set K2
so that
Ã
!1/2
X
⊥ e 2
C(e)kPK
I kR
22|K1 | |K1 |
< ǫ/2 .
2
e∈K1
Then for all A2 ∈ F(E \ K2 ), we have |P[A ∩ A2 ] − P[A]P[A2 ]| < ǫ.
To prove Theorem 8.4, we need to establish some lemmas. Let G = (V, E) be a finite
network and F ⊂ E. Recall that the set of edges of the contracted graph G/F is identified
with E, with each f ∈ F being a loop in G/F . When we consider the graph G \ F left
after deletion of F , the space of functions on E \ F is identified with the space of functions
on E that vanish on F .
F
For every S ⊂ F , let ⋆F
S be the span of the stars in the graph GS where every edge
in S is contracted and every edge in F \ S is deleted.
Suppose that f ∈ F . Note that the space of cycles in G/f is the space spanned by
¡ ¢⊥
the cycles in G and χf . Consequently, the space of stars in G/f is ⋆ ∩ χf . It is also
¡
¢ ¡ ¢⊥
easy to see that the space of stars in G \ f is ⋆ + Rχf ∩ χf . Induction then gives
¡
¢
χF \S i ∩ hχF i⊥ .
⋆F
(8.8)
S = ⋆+h
§8. Ergodic Properties
29
2
Let QF be the (random) orthogonal projection onto the subspace ⋆F
F ∩T of ℓ− (E)
spanned by the stars of GF
F ∩T . We thank Ben Morris for simplying our original proof of
the following lemma.
Lemma 8.5. Let G = (V, E) be a finite network and F ⊂ E. Then
X
EQF =
P[T ∩ F = S]P⋆FS = PF⊥ P⋆ PF⊥ .
S⊆F
(8.9)
P
χei , where hei i is the path in T between
the endpoints of e and the path is oriented so that χe − ζTe ∈ ♦. Then P⋆ χe = EζTe by
Kirchhoff’s Theorem 4.1. Likewise, for e ∈
/ F , we have P⋆FS χe = PF⊥ E[ζTe | T ∩ F = S].
Proof. For a spanning tree T of G, let ζTe :=
i
To prove (8.9), we show that for any e, h ∈ E, we have
E(QF χe , χh )R = (PF⊥ P⋆ PF⊥ χe , χh )R .
(8.10)
If either e or h lies in F , then both sides of (8.10) are 0 by (8.8). Thus, we may suppose
that e, h ∈
/ F . In this case, (8.10) reduces to E(QF χe , χh )R = (P⋆ χe , χh )R . This follows
from conditioning on T ∩ F :
¢
¢
¡
¡
¢
¡
(P⋆ χe , χh )R = EζTe , χh R = EζTe , PF⊥ χh R = PF⊥ EζTe , χh R
X
¯
¢
¡
=
P[T ∩ F = S] PF⊥ E[ζTe ¯ T ∩ F = S], χh R
S⊆F
=
X
S⊆F
¢
¡
P[T ∩ F = S] P⋆FS χe , χh R = E(QF χe , χh )R .
Lemma 8.6. Let G be a finite network, F ⊂ E, and ξ ∈ ℓ2− (E). Then
Var(QF ξ) := EkQF ξ − EQF ξk2R = kPF P⋆ PF⊥ ξk2R .
Proof. Computing the square of the norm by an inner product, we find
EkQF ξ − EQF ξk2R = E(QF ξ − EQF ξ, QF ξ − EQF ξ)R
= E(QF ξ, ξ)R − 2E(QF ξ, EQF ξ)R + kEQF ξk2R
[since QF is an orthogonal projection]
= (EQF ξ, ξ)R − 2kEQF ξk2R + kEQF ξk2R
[by linearity of inner product]
= (PF⊥ P⋆ PF⊥ ξ, ξ)R − kPF⊥ P⋆ PF⊥ ξk2R
= (P⋆ PF⊥ ξ, PF⊥ ξ)R − kPF⊥ P⋆ PF⊥ ξk2R
= kP⋆ PF⊥ ξk2R − kPF⊥ P⋆ PF⊥ ξk2R
= kPF P⋆ PF⊥ ξk2R .
In the next lemma, we use the usual ℓ2 -norm on Rk .
[by (8.9)]
§8. Ergodic Properties
30
Lemma 8.7. Let P be any probability measure on the set of k×k real matrices. For a matrix
M , write its rows as Mi and its entries as Mi,j (i = 1, . . . , k). If E det M = det EM and
each kMi k2 6 1 a.s., then
Var det M 6 k
k
X
Var Mi,j .
i,j=1
Proof. We use the notation M = [M1 , M2 , . . . , Mk ]. Hadamard’s inequality (see, e.g.,
Beckenbach and Bellman (1965)) gives us
¯
¯
k
¯X
¯
¯
¯
| det M − det EM | = ¯
det[EM1 , . . . , EMi−1 , Mi − EMi , Mi+1 , . . . , Mk ]¯
¯
¯
i=1
6
k
X
i=1
6
k
X
i=1
kEM1 k2 · · · kEMi−1 k2 kMi − EMi k2 kMi+1 k2 · · · kMk k2
kMi − EMi k2 .
Therefore
Var det M = E| det M − E det M |2 = E| det M − det EM |2
" k
#
k
X
X
2
6E k
kMi − EMi k2 = k
Var Mi .
i=1
i=1
Proof of Theorem 8.4. By (4.8), we have
P[T ∩ K = B | T ∩ F ] = det MBK ,
where
MBK
with notation as in (4.7). Now
h¡
i
B,e χ̂e χ̂e′ ¢
,
:= QF
R
e,e′ ∈K
χ̂e , χ̂e )R ]e,e′ ∈K
EMBK = [(EQB,e
h¡ F
i
′¢
= (PF⊥ P⋆ PF⊥ )B,e χ̂e , χ̂e R
by (8.9)
e,e′ ∈K
h¡
i
B,e ⊥ χ̂e χ̂e′ ¢
PF ,
= PF⊥ P⋆
R e,e′ ∈K
h¡
i
B,e χ̂e χ̂e′ ¢
since K ∩ F = ∅.
= P⋆
,
R
′
′
e,e ∈K
§8. Ergodic Properties
31
Therefore,
E det MBK = EP[T ∩ K = B | T ∩ F ] = P[T ∩ K = B] = det EMBK .
Furthermore, for any orthogonal projection P , we have
X
′
e′ ∈K
(P χ̂e , χ̂e )2R 6 kP χ̂e k2R 6 1 ,
because hχ̂e : e′ ∈ E1/2 i is an orthonormal basis for ℓ2− (E). Thus, we may apply Lemma 8.7
to obtain
′
¡
¢
¡
¢
Var P[T ∩ K = B | T ∩ F ] = Var det MBK
´
³
X
e χ̂e′
χ̂
,
6 |K|
Var QB,e
F
e,e′ ∈K
6 |K|
= |K|
= |K|
X
e∈K,e′ ∈E
X
e∈K
X
e∈K
R
´
³
e χ̂e′
χ̂
,
Var QB,e
F
¢
¡
e
χ̂
Var QB,e
F
C(e)kPF I e k2R ,
using Lemma 8.6 and (4.7). This proves (8.6).
It is easy to deduce (8.7) from (8.6): Write a := 22|K| |K|
we have
R
¡
¢
Var P[A1 | T ∩ F ] 6 a
P
e∈K
C(e)kPF I e k2R . Then
since A1 is the union of at most 2|K| cylinder events in F(K). Therefore
so that
¯
¯
¯P[A1 | A2 ] − P[A1 ]¯2 P[A2 ] 6 a ,
¯
¯
¯P[A1 ∩ A2 ] − P[A1 ]P[A2 ]¯2 6 aP[A2 ] 6 a .
This is the same as (8.7).
Finally, we remark that in case G is an amenable Cayley graph, Pemantle (2000)
has shown that the uniform spanning forest measure is (strongly) Følner independent, a
mixing property that is still stronger than tail triviality.
§9. The Number of Components
32
§9. The Number of Components.
When is the free spanning forest or the wired spanning forest a single tree, as in the
recurrent case? The following answer for the wired spanning forest is due to Pemantle
(1991):
Proposition 9.1. Let G be any network. The wired spanning forest is a single tree a.s.
iff from every (or some) vertex, random walk and independent loop-erased random walk
intersect infinitely often a.s. Moreover, the probability that u and v belong to the same tree
equals the probability that random walk from u intersects independent loop-erased random
walk from v.
This is obvious from Theorem 5.1 (which wasn’t available to Pemantle at the time).
It turns out that for any transient Markov chain, if two independent copies of the
chain started at any two different states intersect with probability 1, then the first chain
intersects the loop erasure of the second a.s.; see Lyons, Peres, and Schramm (1998). (More
generally, given that two independent chains X and Y with the same law intersect i.o.,
the conditional probability that X intersects the loop erasure of Y i.o. is 1.) This makes
it considerably easier to decide whether the wired spanning forest is a single tree. Thus:
Theorem 9.2. (Connectedness of WSF) Let G be any network. The wired spanning
forest is a single tree a.s. iff two independent copies of the Markov chain corresponding to
G started at any two different states intersect with probability 1.
How many trees are in the wired spanning forest when the condition of this theorem
does not hold? Often infinitely many a.s., but there can be only finitely many:
Example 9.3. Join two copies G1 , G2 of the usual nearest-neighbor graph of Z3 by an edge
e. Let G denote the resulting graph. Any spanning tree of a finite connected subgraph
G′ ⊂ G that intersects G1 and G2 consists of a spanning tree of G1 ∩ G′ , a spanning tree
of G2 ∩ G′ and the edge e. Consequently, the free uniform spanning forest of G is obtained
by appending e to the union of an FSF of G1 and an independent FSF of G2 . Therefore,
the FSF on G is a tree a.s. But the wired uniform spanning forest has two trees a.s. by
Theorem 9.4 below.
To give the general answer to how many trees are in the WSF, we use the following
quantity: let α(w1 , . . . , wK ) be the probability that independent random walks started at
w1 , . . . , wK have no pairwise intersections.
Theorem 9.4. Let G be a connected network. The number of trees of the WSF is a.s.
©
ª
sup K : ∃w1 , . . . , wK α(w1 , . . . , wK ) > 0 .
(9.1)
§9. The Number of Components
33
Moreover, if the probability is 0 that two independent random walks from every (or some)
vertex v intersect infinitely often, then the number of trees of the WSF is a.s. infinite.
In particular, the number of trees of the WSF is equal a.s. to a constant. The case
of the free spanning forest (when it differs from the wired) is largely mysterious. In
particular, we do not know whether the number of components is deterministic or random
(Question 15.7). See Theorem 12.7 for one case that is understood.
Proof. Let hXv (n)iv∈V be a collection of independent random walks indexed by their ini-
tial states. First suppose that α(w1 , . . . , wK ) > 0. Then by Lévy’s 0-1 law, for every
¡
¢
ǫ > 0, there is an n ∈ N such that α Xw1 (n), . . . , XwK (n) > 1 − ǫ with probability
′
′
> α(w1 , . . . , wK )/2. In particular, there are w1′ , . . . , wK
such that α(w1′ , . . . , wK
) > 1 − ǫ.
′
′
Using Wilson’s method rooted at infinity starting with the vertices w 1 , . . . , wK , this im-
plies that with probability greater than 1 − ǫ, the number of trees for WSF is at least K.
As ǫ > 0 was arbitrary, this implies that the number of WSF trees is a.s. at least (9.1).
For the converse, suppose that with positive probability, the number of trees in the
WSF is at least k. Then there are vertices w1 , . . . , wk such that the event that they belong
to k different components of the WSF has positive probability. We claim that with positive
¡
¢
probability, α Xw1 (n), . . . , Xwk (n) → 1 as n → ∞. For if not, then by Lévy’s 0-1 law
again, there would a.s. exist i 6= j with infinitely many intersections between hX wi (n)i and
hXwj (n)i, whence also between hXwi (n)i and LEhXwj (n)i by Lyons, Peres, and Schramm
(1998). But then, by Wilson’s method, the probability that wi and wj belong to the same
tree for some i 6= j would be 1, contradicting our assumption. This proves the claim and
that the number of trees is WSF-a.s. at most (9.1).
Moreover, if the probability is zero that two independent random walks X 1 , X 2 inter¡
¢
sect i.o. starting at some w ∈ V, then limn→∞ α X 1 (n), X 2 (n) = 1 a.s. Therefore
¡
¢
lim α X 1 (n), . . . , X k (n) = 1
n→∞
a.s. for any independent random walks X 1 , . . . , X k . This implies that the number of
components of WSF is a.s. infinite.
Remark 9.5. If the number of components of the WSF is finite a.s., then it a.s. equals the
dimension of the vector space BH(G) of bounded harmonic functions on G. [Note that
when HD(G) 6∼
= R, then also BH(G) ∩ HD(G) 6∼
= R; see, e.g., Soardi (1994), Thm. 3.73.]
Indeed, suppose that there are k components in the WSF and that v1 , . . . , vk are vertices
satisfying α(v1 , . . . , vk ) > 0. Let {Xvi : 1 6 i 6 k} be independent random walks indexed
by their initial states. Consider the random functions
hi (w) := P[Xw′ intersects Xvi i.o. | Xv1 , . . . , Xvk ] ,
§9. The Number of Components
34
where the random walk Xw′ starts at w and is independent of all Xvi . Then it can be
shown that a.s. on the event that Xv1 , . . . , Xvk have pairwise disjoint paths, the functions
{h1 , . . . , hk } form a basis for BH(G).
Corollary 9.6. (The Phase Transition at Dimension 4) Let G be a transitive
graph. Denote by B(o, n) the ball of radius n centered at the identity o. If |B(o, n)| = O(n 4 )
as n → ∞ (e.g., if G = Zd for d 6 4), then the WSF on G has one tree a.s. On the other
hand, if |B(o, n)|/n4 → ∞ (e.g., if G = Zd for d > 5), then the WSF on G has infinitely
many trees a.s.
Proof. As explained in Lyons, Peres and Schramm (1998), from the known asymptotics
of the Green function in transitive graphs, it easily follows that two independent simple
random walks in G have infinitely many intersections a.s. if |B(o, n)| = O(n 4 ) as n → ∞
and finitely many intersections a.s. otherwise (see Lawler (1991) for the case of Z d ). The
theorem now follows from Theorems 9.2 and 9.4.
Remark 9.7. Benjamini, Kesten, Peres, and Schramm (1998) give the following result
concerning the relative placement of the trees in the uniform spanning forest in Z d . Identify
each tree in the uniform spanning forest on Zd to a single point. In the induced metric,
the diameter of the resulting (locally infinite) graph is a.s. ⌊(d − 1)/4⌋.
Remark 9.8. (The Number of FSF Components) Let NF = NF (G) be the number
of components of the FSF in a network G. If Aut(G) has an infinite orbit, then ergodicity
(Corollary 8.2) shows that NF (G) is a.s. constant. For general G, we have FSF(NF <
∞) ∈ {0, 1} by tail triviality (Theorem 8.3). To illustrate that N F may be finite and
larger than 1, we provide the following example: Let G0 be formed by two copies of Z3
joined by an edge [x, y]; put G := G0 × Z. By Theorem 7.3 and the result of Thomassen
mentioned before it, FSF(G) = WSF(G). Transience of Z3 implies that with positive
probability, independent random walks in G0 started at x and y have disjoint paths. Since
simple random walk in G projects on G0 to a (delayed) simple random walk, it follows
that NF (G) > 2 a.s. by Theorem 9.4. But by Pemantle’s theorem (Corollary 9.6 for Z 4 ),
NF (G) 6 2 a.s.
Example 9.9. (Free Product) Let G be the Cayley graph of the free product Z d ∗ Z2 ,
where Z2 is the group with two elements, with the obvious generating set. Then G is
transitive. Note that the removal of any Z2 edge in G separates G into two infinite
components. It follows that the FSF on G is obtained by taking independent FSF’s on
each Zd copy lying in G, then adding the Z2 edges. Therefore, the FSF is connected iff
d 6 4, and FSF 6= WSF for all d > 0.
§10. Ends of WSF Components in Transitive Graphs
35
§10. Ends of WSF Components in Transitive Graphs.
In this section, we complete and extend Pemantle’s (1991) theorem on the number of
ends in spanning forests. The conclusion of Theorems 10.3, 10.4 and 10.6, and Proposition 10.10 will be:
Theorem 10.1. If G is network with a transitive unimodular automorphism group, then
each tree of the wired spanning forest on G has almost surely one end unless G is roughly
isometric to Z, in which case it has two ends a.s.
Of course, all (discrete) countable groups are unimodular, whence all Cayley graphs
have a transitive unimodular group action.
Possible extensions of Theorem 10.1 are proposed in Questions 15.3, 15.4, and 15.5.
We begin the proof of Theorem 10.1 with the simple result that there are at most 2
ends per tree:
Lemma 10.2. If G is a network with a transitive unimodular automorphism group, then
each tree of the wired spanning forest on G has almost surely at most 2 ends.
Proof. This is immediate from Thm. 7.2 of BLPS (1999) in conjunction with Theorem 6.4.
We now discuss the case where the network is transient and the forest is a single tree:
Theorem 10.3. If G is a transitive transient network and the WSF on G is a single tree
a.s., then that tree has a single end a.s.
Proof. If the tree T has 2 ends, then there is a unique bi-infinite path that does not
backtrack such that one direction gives one end of T and the other direction gives the
other end. Call this bi-infinite path the trunk of the tree. Let p be the probability that a
vertex is on the trunk, which is the same for all vertices by transitivity. We want to show
that p = 0.
Our argument is inspired by the proof of Thm. 4.3 of Pemantle (1991). By tail
triviality, the probability that x and y are both on the trunk tends to p 2 as their distance
tends to infinity. Use Wilson’s method rooted at infinity, starting with the vertex x, then y.
In order that x, y ∈ trunk, it is necessary that the loop-erased random walk from x contains
y or that the random walk from y first hits this loop-erased path at x. Consequently,
P[x, y ∈ trunk] 6 Px [τy < ∞] + Py [τx < ∞] .
Note that Px [τy < ∞] Py [τx < ∞] is a lower bound for the probability that for some
n > 2 dist(x, y), the random walk starting at x will be at x again at time n. Accordingly,
§10. Ends of WSF Components in Transitive Graphs
36
transience implies that Px [τy < ∞] Py [τx < ∞] → 0 as dist(x, y) → ∞. Now by (6.2), we
have Px [τy < ∞] = Py [τx < ∞]. Hence,
p2 =
lim
dist(x,y)→∞
P[x, y ∈ trunk] 6 2
lim
dist(x,y)→∞
Px [τy < ∞] = 0 ,
which gives p = 0, as desired.
Next, we deal with transient networks having a disconnected spanning forest.
Theorem 10.4. Let G be a network with a transitive unimodular automorphism group,
and assume that with positive probability the WSF is disconnected. Then WSF-a.s., every
tree has only one end.
Proof. By ergodicity (Section 8), we know that the WSF is a.s. disconnected. Let F be the
wired spanning forest. We have seen that a.s. each component of F has one or two ends.
Let Γ be a transitive unimodular automorphism group of G.
Let T be a component of F. Define trunk(T ) to be the trunk of T as in the proof of
Theorem 10.3 if T has 2 ends, else the empty set. We need to show that the trunk of each
component of F is empty a.s.
Choose a basepoint o ∈ V. Let Ao be the event that o is in the trunk of its component
To , and A′o be the event that trunk(To ) 6= ∅. If there is positive probability that a
component of F has two ends, then P[A′o ] > P[Ao ] > 0. Aiming for a contradiction, then,
assume that P[Ao ] > 0.
For every vertex v ∈ trunk(To ), let the bush of v be the set of vertices w in To
such that w is in a finite component of To \ v. Conditioned on A′o , for every vertex w in
To \ trunk(To ), there is precisely one vertex v ∈ trunk(To ) such that w is in the bush of v.
Let Bo be the (possibly empty) bush of o.
We claim that
¯ ¤
£
E |Bo | ¯ Ao < ∞ .
(10.1)
To verify this, for vertices v, w ∈ V, let g(v, w) be the probability that w is in the bush
of v. Clearly, g(•, •) is invariant under the diagonal action of Γ. By the Mass-Transport
Principle (see BLPS (1999), Section 3),
X
z∈V
g(o, z) =
X
g(z, o) .
z∈V
On the left is the expected size of Bo , while on the right is the expected number of z whose
bush contains o, i.e., the probability that o is in a bush. Since this is at most 1, the result
follows.
§10. Ends of WSF Components in Transitive Graphs
37
Let F(trunk, o) denote the σ-field of events depending only on trunk(To ). Let Po [ • ] :=
P[ • | Ao ] be the probability distribution of the OWSF conditioned on Ao . The following
lemma identifies the distribution of Po [ • | F(trunk, o)].
Suppose that ω is a subgraph of G. We use Wilson’s method rooted at {∞} ∪ ω;
that is, set F (0) := ω; inductively, at step n + 1, start a random walk at a new vertex,
stopping if F (n) is hit, and set F (n + 1) to be F (n) union with the loop erasure of that
walk. We make sure that all vertices are eventually visited, and hence, by the proof of
S
Wilson’s theorem, the distribution of n F (n) is independent of the order in which the
S
vertices starting the walks are chosen. Let νω be the law of n F (n).
Lemma 10.5. (Trunk Lemma) Assuming that P[Ao ] > 0, for every event D, almost
surely,
Po [D | F(trunk, o)] = νtrunk(To ) (D) .
(10.2)
Proof. We first consider the case where G is a (right) Cayley graph of Γ. In that case, we
take o to be the identity of Γ for simplicity. For every x ∈ V, there is a unique γ x ∈ Γ
satisfying γx o = x; in fact, γx v = xv.
In the OWSF, there is exactly one oriented edge in F leading out of every vertex x;
−1
let s(x) be the other endpoint. Write S(F) for γs(o)
F. We claim that the restriction of S
to Ao is measure preserving; that is,
Po = SPo .
(10.3)
Here, SPo is the probability measure given by SPo [D] = Po [S −1 D].
To verify (10.3), let D be an event. For x, y ∈ V, let ϕ(x, y) be the OWSF-probability
of the event {y = s(x)} ∩ Ax ∩ γy D, where Ax := γx Ao is the event that x is in the trunk
of its OWSF-component. Clearly, ϕ is invariant under the diagonal action of Γ, whence
the Mass-Transport Principle implies that
X
ϕ(x, o) =
x
X
ϕ(o, y) .
y
Note that
[
x
{o = s(x)} ∩ Ax ∩ γo D =
[
x
{o = s(x)} ∩ Ax ∩ D = Ao ∩ D
and the union is disjoint (up to a set of zero measure), while
[
y
{y = s(o)} ∩ Ao ∩ γy D = Ao ∩ S −1 D
(10.4)
§10. Ends of WSF Components in Transitive Graphs
38
and the union is disjoint. Consequently, (10.4) gives OWSF[A o ∩ D] = OWSF[Ao ∩ S −1 D],
which implies (10.3).
Given ǫ > 0, let Kǫ be a cylinder event depending on edges in a finite set Bǫ such
that OWSF[Kǫ △Ao ] < ǫ. Let µǫ be OWSF conditioned on Kǫ and let Vǫ be all the vertices
incident with Bǫ .
Let k•k denote the total variation norm. It is easy to see that there is a constant c
depending only on P[Ao ] such that kPo − µǫ k 6 cǫ. Consequently,
kPo − S n µǫ k = kS n (Po − µǫ )k 6 kPo − µǫ k 6 cǫ .
(10.5)
Let Ω̂ be the measurable space of pairs of configurations (F, ω) where F and ω are sets
of oriented edges of G. The configurations we shall be considering are those (F, ω) where
F is an oriented spanning forest of G.
We now describe a measure µ̂ǫ on Ω̂ such that the projection of µ̂ǫ on the first coordinate gives µǫ . Use Wilson’s method rooted at ∞, but starting only from the vertices in
Vǫ . Let ω be the resulting forest obtained and condition on ω ∈ Kǫ . Now continue with
Wilson’s method, visiting all vertices of G, and let F be the resulting OWSF. Define µ̂ ǫ to
be the law of (F, ω), and note that, indeed, the projection of µ̂ǫ on the first coordinate is
µǫ . Also note that conditioned on ω the µ̂ǫ law of F is νω .
Here is an alternative way to describe µ̂ǫ . Given a vertex v and given F, define the
future of v, fu(v) = fuF (v), to be the oriented path hv, s(v), s(s(v)), . . .i in F. For a set of
S
vertices W ⊂ V, define fu(W ) := v∈W fu(v). Then µ̂ǫ is the image of OWSF conditioned
¡
¢
on Kǫ under the map Φǫ (F) := F, fuF (Vǫ ) .
We define S on Ω̂ by shifting both coordinates; that is, if x = s(o) in F, set S(F, ω) :=
(γx−1 F, γx−1 ω).
Take some sequence nk → ∞ such that S nk µ̂ǫ has a weak limit µ̂∞
ǫ as k → ∞. Observe
that also for µ̂∞
ǫ , when we condition on ω, F is given by νω .
Let ηǫ be the image of Po under the map Φǫ , and let ηǫ∞ be a weak limit of S nk ηǫ .
Then it easily follows from (10.5) that
kηǫ∞ − µ̂∞
ǫ k 6 cǫ .
¡
¢
Let η ′ be the image of Po under the map F 7→ F, fu(o) . Note that for any two fixed
vertices v and u, the probability that fu(v) \ fu(u) intersects a ball of fixed radius about
sn (u) tends to 0 as n → ∞. It follows that ηǫ∞ is also a weak limit of S nk η ′ , because
¡
¢
S n fu(Vǫ ) \ fu(o) tends a.s. to the empty set. Let η be the image of Po under the map
¡
¢
F 7→ F, trunk(To ) . Note that η = limn S n η ′ by (10.3). Hence, we have
kη − µ̂∞
ǫ k 6 cǫ .
(10.6)
§10. Ends of WSF Components in Transitive Graphs
39
Since ǫ > 0 is arbitrary and µ̂∞
ǫ conditioned on ω is given by νω , the same is true for η,
which gives (10.2).
It remains to generalize to the case where G is not a Cayley graph of Γ. For each
x ∈ V, let γx be some automorphism in Γ taking o to x. The problem is that there is no
canonical choice of γx , and hence ϕ as defined above is not invariant under the diagonal
action of Γ. However, the same proof as above does show that
Po (D) = SPo (D)
holds for every event D that is invariant under the stabilizer Γo of o in Γ. Using a simple
averaging argument, it is not hard to see that the cylinder event Kǫ can be chosen to be
Γo -invariant. By following the same arguments as in the proof above, one concludes that
(10.6) holds when the measures there are restricted to the σ-field of Γo -invariant events,
where Γo acts diagonally on Ω̂. Given a measure µ, let Γo µ denote the measure given by
Z
µ(γD) dγ ,
Γo µ(D) :=
γ∈Γo
where the integration is with respect to Haar measure normalized to give Γ o measure 1.
Since (10.6) holds when the measures are restricted to the σ-field of Γo -invariant events
and Γo η = η, it follows that
kη − Γo µ̂∞
ǫ k 6 cǫ .
Then the argument is completed as above.
We resume the proof of Theorem 10.4. By Lemma 10.5 and our assumption that a.s. F
is disconnected, it follows that there is a.s. some vertex w ∈ V such that P ow [τtrunk(To ) = ∞ |
F(trunk, o)] > 0. Let W be the union of connected components of G \ trunk(T o ) containing
such vertices. Then for each w such that Po [w ∈ W ] > 0, we have a.s. Pow [τtrunk(To ) = ∞ |
F(trunk, o), w ∈ W ] > 0. Now Po -a.s., there is a vertex v ∈ trunk(To ) that neighbors with
some vertex in W . Hence, Po [o ∈ ∂V W | F(trunk, o)] > 0 and therefore
+
Poo [τtrunk(T
= ∞] > 0 ,
o)
(10.7)
because the first step of the random walk starting at o may be to some vertex in W , and
then, with positive probability, the random walk never visits trunk(T o ) again.
Now Lemma 10.5 shows that Bo , the bush of o, satisfies
£
¤
Po [w ∈ Bo | F(trunk, o)] = Pow τo < τtrunk(To )\{o} | F(trunk, o)
¤
£
> Pow τo < τtrunk(To )\{o} ∧ τw+ | F(trunk, o)
h
i
+
|
F(trunk,
o)
= Poo τw < τtrunk(T
o)
§10. Ends of WSF Components in Transitive Graphs
40
by reversibility and transitivity. Consequently, the expected size of B o conditioned on
trunk(To ) and Ao is bounded below by the expected number of vertices visited by a random
walk started at o before returning to trunk(To ). However, (10.7) says that there is positive
probability that a random walk started at o never comes back to trunk(T o ). Therefore, the
expected number of vertices in Bo conditioned on Ao is infinite. This contradicts (10.1).
In this proof, we have looked at the bushes of the (nonexistent) trunk. However, one
can also try to study the bushes of the ray fu(o). See Conjecture 15.12.
Finally, we deal with the recurrent case.
Theorem 10.6. Let G be a recurrent transitive network, and suppose that T has two ends
with positive probability. Then G is roughly isometric to Z.
Remark 10.7. If G is a Cayley graph of a group Γ and G is roughly isometric to Z, then
Γ is a finite extension of Z. This follows from Gromov’s (1981) classification of groups of
polynomial growth.
Lemma 10.8. If G is a transitive graph with two ends, then G is roughly isometric to Z.
See Propn. 6.1 of Mohar (1991). One can also obtain a rough isometry f : G → Z as
follows: Consider a finite connected set K such that G \ K has two infinite components
C1 , C2 . Define f (v) := dist(K, v) for v ∈ C1 and f (v) := −dist(K, v) for v ∈ C2 . It is not
hard to check that f is a rough isometry from G to Z.
Proof of Theorem 10.6. The theorem follows immediately from Lemma 10.8 and the following lemma.
Lemma 10.9. The assumptions of Theorem 10.6 imply that G has two ends.
Proof. Recall that the trunk of a tree with two ends is its unique bi-infinite simple path.
By ergodicity, it follows that T has two ends a.s., and therefore a unique trunk a.s.
As the details of the proof are somewhat tedious, we begin with a sketch. We show
that the two ends of T are representatives of two ends of G. Pick three points a, b, c far
away from each other. With high probability, the path in T from each of these points to
the trunk is not too long. See Fig. 1. Since the trunk is isomorphic as a graph to Z, it
is meaningful to say that a segment of the trunk is between two vertices on the trunk.
Consider the case where the part of the trunk close to a is between the part close to b and
the part close to c. Let x be a point whose distance from a is large, but much smaller than
the distances from a to c or from a to b. With high probability, the trunk also passes not
far from x. Using Wilson’s method rooted at a, it follows that either (1) with probability
§10. Ends of WSF Components in Transitive Graphs
41
bounded away from 0, the loop erasure of a random walk from b to a passes near x before
hitting a, or (2) with probability bounded away from 0, a random walk starting from c
passes near x before hitting a. However, if both these probabilities are bounded away from
zero, then the meeting point y of the tree paths from c to a and from b to a has a good
probability of being far from a. That contradicts the assumption that the part of the trunk
close to a is between the parts of the trunk close to b and to c. It follows that each such x
is likely to be close either to the random walk from c to a or to the tree path from b to a,
but not both. That partitions the set of such x’s into two subsets that do not share edges.
In the limit, as b and c drift far away, we see that G has more than one end.
to c
trunk
a
b
y
K
Figure 1. The trunk passes close to a, b, and c.
We now begin the actual proof. Fix some very small ǫ > 0. For every v ∈ V and
d > 0, let Adv be the event that the tree path from v to the trunk is contained in the ball
B(v, d) centered at v of radius d. Note that limd→∞ P[Adv ] = 1. Let d0 be sufficiently large
that
P[Adv0 ] > 1 − ǫ
(10.8)
for all v ∈ V. To avoid clutter, we write Av instead of Adv0 . Let r0 be much larger than
d0 and take any r1 > r0 . Fix a vertex a ∈ V and let b ∈ V be some vertex with dist(a, b)
much larger than r1 . (Note that dist denotes distance in G, not in the tree.) Let c be
some vertex with dist(a, c) much larger than dist(a, b). Let H be the event that a network
random walk starting at c will hit a before b. By interchanging a and b if necessary, assume
without loss of generality that
P[H] > 1/2 .
(10.9)
§10. Ends of WSF Components in Transitive Graphs
42
We assume that dist(a, c) is so much larger than dist(a, b) that
Pb [B(c, d0 + 1) is hit before a] < ǫ .
(10.10)
Let Pb be the loop erasure of the random walk starting at b and stopped at a. By
using Wilson’s method with root a and starting vertex b, we may take P b ⊂ T . Now let Xc
be an independent random walk starting at c and stopped at a, and let Z be the image of
Xc . Let τ be the first time t that Xc (t) ∈ Pb , and set y := Xc (τ ). Note that by Wilson’s
method, we may take the loop erasure of Xc restricted to [0, τ ] to be the tree path from c
to Pb . Hence, y is the “meeting point” of a, b, c in the tree.
Provided that dist(a, b) is sufficiently large, we have
£ ¯
¤
P H ¯ y ∈ B(b, d0 + 1) < ǫ ,
(10.11)
because the probability that a will be hit before b by a random walk starting at y ∈
B(b, d0 + 1) tends to zero as dist(a, b) → ∞.
For any pair v, w ∈ V, on the event Av ∩ Aw , the tree path joining v to w is contained
in trunk ∪ B(v, d0 ) ∪ B(w, d0 ). Consider the three tree paths joining y to a, b and c.
These paths are disjoint, with the exception of y. Hence there is at least one neighbor
of y that is on one of these paths but not on the trunk. On the event Aa ∩ Ab ∩ Ac ,
these three paths are contained in trunk ∪ B(a, d0 ) ∪ B(b, d0 ) ∪ B(c, d0 ), and therefore
y ∈ B(a, d0 + 1) ∪ B(b, d0 + 1) ∪ B(c, d0 + 1). As P[Av ] > 1 − ǫ for every v, we obtain
£
¤
P y ∈ B(a, d0 + 1) ∪ B(b, d0 + 1) ∪ B(c, d0 + 1) > 1 − 3ǫ .
From (10.10), we have
whence the preceding gives
£
¤
P y ∈ B(c, d0 + 1) < ǫ ,
£
¤
P y ∈ B(a, d0 + 1) ∪ B(b, d0 + 1) > 1 − 4ǫ .
(10.12)
However, assuming that dist(a, b) is sufficiently large, by (10.11), we have
£
¤
£ ¯
¤
P y ∈ B(b, d0 + 1), H 6 P H ¯ y ∈ B(b, d0 + 1) < ǫ ,
(10.13)
which by (10.12) and (10.9), implies that
£
¤
P y ∈ B(a, d0 + 1), H > 1/2 − 5ǫ .
(10.14)
§10. Ends of WSF Components in Transitive Graphs
43
Moreover, by (10.13) and (10.12),
¯ ¤
¯ ¤
£
£
P y ∈ B(a, d0 + 1) ¯ H = P y ∈ B(a, d0 + 1) ∪ B(b, d0 + 1) ¯ H
¯ ¤
£
− P y ∈ B(b, d0 + 1) ¯ H
£
¤
>1−P y ∈
/ B(a, d0 + 1) ∪ B(b, d0 + 1) /P[H]
£
¤
− P y ∈ B(b, d0 + 1), H /P[H]
> 1 − (4ǫ + ǫ)/P[H]
> 1 − 10ǫ .
(10.15)
Set K := B(a, r1 ) \ B(a, r0 ). For x ∈ K, let
Let Bx be the event
£
¤
f (x) := P dist(x, Pb ) < d0 ,
£
¤
g(x) := P dist(x, Z) < d0 , H .
´
©
ª ³©
ª
Bx := dist(x, Pb ) < d0 ∪ dist(x, Z) < d0 ∩ H .
Fix an x ∈ K. As dist(x, a) 6 r1 , by assuming that dist(a, b) is much larger than r1 , we
can make the probability that the trunk (with either orientation) will visit B(a, d 0 + 1),
B(b, d0 ) and B(x, d0 ) in that order be smaller than ǫ; in other words, with probability at
most ǫ, there is a parameterization φ : Z → trunk of the trunk and integers i a < 0 < ix
such that φ(0) ∈ B(b, d0 ), φ(ia ) ∈ B(a, d0 + 1), and φ(ix ) ∈ B(x, d0 ). A similar statement
applies with c replacing b. Consequently, if x, a, b, c are all near the trunk and y is near
a, then with probability at least 1 − ǫ, the point x is near the tree path joining a and c or
near the tree path joining a and b; that is,
¸
·³
´
©
ª
P
y ∈ B(a, d0 + 1) ∩ Aa ∩ Ab ∩ Ac ∩ Ax \ Bx 6 ǫ .
By the definition of f and g and by (10.14) and (10.8), it follows that
f (x) + g(x) > P[Bx ]
¸
·
©
ª
> P y ∈ B(a, d0 + 1) ∩ Aa ∩ Ab ∩ Ac ∩ Ax
·³
¸
´
©
ª
−P
y ∈ B(a, d0 + 1) ∩ Aa ∩ Ab ∩ Ac ∩ Ax \ Bx
> (1/2) − 5ǫ − 4ǫ − ǫ = (1/2) − 10ǫ .
(10.16)
§10. Ends of WSF Components in Transitive Graphs
44
ª
©
ª
Conditioned on the event {dist(x, Z) < d0 ∩ H ∩ dist(x, Pb ) < d0 , provided that
r0 is sufficiently larger than d0 , the probability that y ∈ B(a, d0 + 1) is smaller than ǫ,
because a random walk starting at some point close to Pb ∩ B(x, d0 ) is unlikely to get as
far as B(a, d0 +1) before hitting Pb ∩B(x, d0 ), while a random walk starting in B(a, d0 +1)
is unlikely to get out of B(a, r0 ) before hitting a. Hence
¯
i
h
¯
1−ǫ6P y ∈
/ B(a, d0 + 1) ¯ dist(x, Z) < d0 , H, dist(x, Pb ) < d0
£
¤
P y∈
/ B(a, d0 + 1), H
i
6 h
P dist(x, Z) < d0 , H, dist(x, Pb ) < d0
¯ ¤
£
P y∈
/ B(a, d0 + 1) ¯ H
i
6 h
P dist(x, Z) < d0 , H, dist(x, Pb ) < d0
6
10ǫ
h
i
P dist(x, Z) < d0 , H, dist(x, Pb ) < d0
(10.17)
by (10.15). Note that the event {dist(x, Pb ) < d0 } is independent of each of the events
{dist(x, Z) < d0 } and H. Therefore, by (10.17),
h
i
f (x)g(x) = P dist(x, Z) < d0 , H, dist(x, Pb ) < d0 6 10ǫ/(1 − ǫ) 6 11ǫ
provided ǫ is sufficiently small. Clearly, there is a constant c such that g(x)/g(x ′ ) < c if
x, x′ ∈ K are neighbors. Consequently,
f (x)g(x′ ) 6 11cǫ
(10.18)
if x, x′ ∈ K are neighbors or x = x′ . Set
Kf := {x ∈ K : f (x) > 1/5} ,
Kg := {x ∈ K : g(x) > 1/5} .
It follows from (10.16) and (10.18) that K = Kf ∪ Kg , that Kf and Kg are disjoint, and
that there is no edge connecting them, provided that ǫ is sufficiently small.
Let K ′ be the union of the components of K that have a neighbor in B(a, r0 ) and a
neighbor outside of B(a, r1 ).
We claim that K ′ ∩ Kf 6= ∅ and K ′ ∩ Kg 6= ∅. Note that Pb must intersect K ′ . If we
condition on Pb ∩ Kg 6= ∅, then by an argument similar to the one proving (10.18), there
©
ª
would be probability
at
least
1/5
−
O(ǫ)
that
H
∩
y
∈
/
B(a,
d
+
1)
. However, (10.15)
0
h
©
ªi
6 10ǫ. Therefore, there is positive probability
shows that P H ∩ y ∈
/ B(a, d0 + 1)
(in fact, probability close to one) that Pb ∩ K ′ ∩ Kg = ∅. Because Pb intersects K ′ , this
§10. Ends of WSF Components in Transitive Graphs
45
implies K ′ ∩Kf 6= ∅. An entirely similar argument shows that conditioned on H, with high
probability, the loop erasure of Xc intersected with K is disjoint from Kf . Consequently,
K ′ ∩ Kg 6= ∅.
As Kf and Kg are each nonempty unions of components of K ′ , there are vertices
v, u ∈ K ′ that neighbor with vertices in B(a, r0 ) but that cannot be connected by a path
in B(a, r1 ) − B(a, r0 ). Since r1 may be arbitrarily large, it follows that there are such v, u
that are also in distinct infinite components of G − B(a, r0 ). This means that G has more
than one end.
To complete the proof of Theorem 10.1, we need to show that a transitive graph G
that is roughly isometric to Z has a.s. two ends in its WSF. This follows immediately from
recurrence (so that the WSF is a single tree) and Lemma 10.2.
In fact, it is true even without the assumption that G is transitive:
Proposition 10.10. Let (G, C) be a network with 0 < inf e C(e) 6 supe C(e) < ∞. If G
is roughly isometric to Z and has bounded degrees, then the WSF of G has two ends a.s.
Proof. Note that (G, C) is recurrent, and therefore the WSF is connected a.s. It follows
that a.s. the WSF has at least two ends.
Fix o ∈ V. Take N to be a large integer. Let S be the set of vertices at distance N
from o. It is easy to show that the size of S is bounded independently of N . There is
a partition S = S− ∪ S+ such that the diameter of each of the sets S− , S+ is bounded
independently of N . Use Wilson’s method with root o, starting with the vertices in S +
in any order. Let P be the first path from S+ to o constructed by the method. There
are constants k and c such that there are at least cN disjoint connected subgraphs of size
k that separate S+ from o. Consequently, the probability that a random walk from any
vertex in S+ that stops at o will not hit P decays exponentially with N . That means that
the probability that the WSF contains two paths from o to S+ that are disjoint except at
o tends to zero as N → ∞. Since the same is true for S− , a.s. the tree does not contain
three infinite rays starting at o that are disjoint except at o. Because this is true for any
o, there are at most two ends in the WSF.
Recall that independent (bond) percolation on a graph G may be defined as a
random spanning subgraph ω of G where each edge of G is included in ω independently.
When the inclusion probability for each edge is p, we refer to Bernoulli(p) percolation.
The critical probability pc (G) of a graph G is the supremum of p ∈ [0, 1] for which
Bernoulli(p) percolation has only finite components a.s.
In contrast to Theorem 10.1, we have:
§11. Analysis of the WSF on a Tree
46
Proposition 10.11. If G is a transitive network whose automorphism group is unimodular
and WSF 6= FSF, then FSF-a.s., there is a tree with uncountably many ends, in fact, with
pc < 1.
We do not know if a.s. every FSF-component has infinitely many ends under the above
hypotheses (Question 15.8).
Proof. We have that EWSF [degF x] = 2 for all x, whence EFSF [degF x] > 2 for all x by
Proposition 5.10. Apply Thm. 7.2 of BLPS (1999) and ergodicity.
§11. Analysis of the WSF on a Tree.
In this section, we study the WSF-components on a tree, where a more detailed analysis
is possible. We give a simple derivation of Häggström’s complete description for regular
trees; we give a necessary and sufficient condition for all components of the WSF on an
arbitrary tree to have one end each a.s.; and we specialize to the WSF on spherically
symmetric trees. Also, we prove that all components are recurrent unless the tree contains
a transient ray; this solves a special case of Conjecture 15.1. Since the WSF of a recurrent
tree is the whole tree, it follows that any recurrent tree can arise as a component of WSF.
Consider first the WSF on a regular tree of degree d + 1. Choose a vertex, o, and
begin Wilson’s method rooted at infinity from o. We obtain a ray ξ from o to start our
forest. Now o has d other neighbors, x1 , . . . , xd . By beginning random walks at each of
them in turn, we see that the events Ai := {xi connected to o} are independent given
ξ. Furthermore, it is easy to verify that the probability of a random walk starting at a
neighbor of o ever to visit o is 1/d, so P[Ai | ξ] = 1/d. On the event Ai , we add only
the edge (o, xi ) to the forest and then we repeat the analysis from xi . Thus, the tree
containing o includes, apart from the ray ξ, a critical Galton-Watson tree with binomial
offspring distribution (d, 1/d). In addition, each vertex on ξ has another random subtree
attached to it; its first generation has binomial distribution (d − 1, 1/d), but subsequent
generations yield Galton-Watson trees with binomial distribution (d, 1/d). In particular,
a.s. every tree added to ξ is finite. This means that the tree containing o has only one end,
the equivalence class of ξ. This analysis is easily extended to form a complete description
of the entire wired spanning forest. The resulting description is due to Häggström (1998),
whose work predates Wilson’s algorithm. The WSF-component of the root coincides in
this case with the incipient infinite cluster of the root; see Kesten (1986). For further
information about the component of the root, see Remark 13.3.
§11. Analysis of the WSF on a Tree
47
In general, we see that if we begin Wilson’s method rooted at infinity at a vertex o in
a transient graph, it immediately generates one end of the tree containing o. In order for
this tree to have more than one end, a succession of “coincidences” need to occur, building
up other ends by gradually adding on finite pieces. This is possible (see Corollary 11.4),
but not on transient Cayley graphs (see Theorem 10.1).
A Borel probability measure on the boundary ∂T of a tree T with root o can be
identified with a nonnegative function µ on the vertices of T such that µ(o) = 1 and for
each vertex v, the sum of µ(w) over all children w of v equals µ(v). Denote by M(∂T ) the
collection of such functions µ.
Theorem 11.1. (WSF on General Trees) Let (T, C) be a transient network whose
underlying graph, T , is a tree. Denote h(v) := Pv [τo < ∞]. For any vertex v 6= o, let v̂ be
the parent of v.
(a) If for all µ ∈ M(∂T ), the sum
X
v6=o
µ(v)2 [h(v)−1 − h(v̂)−1 ]
(11.1)
diverges, then all components of the WSF on T have one end a.s.; if this sum converges
for some µ ∈ M(∂T ), then a.s. the WSF on T has components with more than one
end. Furthermore, if (11.1) converges for some µ ∈ M(∂T ), and h(v) → 0 as v → ∞,
then a.s. the WSF on T has components with uncountably many ends.
(b) If for every infinite path ξ in T , the sum of the edge resistances on ξ diverges, then
a.s. all components of the WSF on T are recurrent for the given resistances.
For the proof, we use a general independent percolation ω on T , in which the events
e ∈ ω (e ∈ E) are independent, but may have different probabilities. The following criterion
of Lyons (1992) will be used several times in the course of the proof. See also Lyons and
Peres (1997) for more background.
Theorem 11.2. (Transience-Percolation Criterion) Let (T, C) be an infinite net-
work whose underlying graph is a tree with root o. Given a vertex v ∈ V, let R(o ↔ v)
denote the resistance from o to v, that is, the sum of R(e) = C(e) −1 over the edges e
leading from o to v. Suppose that ω is a general independent percolation on T satisfying
£
¤ 1 + R(o ↔ v̂)
P [v̂, v] ∈ ω =
1 + R(o ↔ v)
for all v ∈ V \ {o}, so that the probability that ω contains the path from o to v is
P[o ↔ v in ω] =
1
.
1 + R(o ↔ v)
§11. Analysis of the WSF on a Tree
48
Then the following are equivalent:
• The network (T, C) is transient.
• There exists µ ∈ M(∂T ) such that
X
v6=o
i
h
µ(v)2 P[o ↔ v in ω]−1 − P[o ↔ v̂ in ω]−1 < ∞ .
• The subgraph ω has infinite components a.s.
We also need the following variant of Lemma 4.2 of Pemantle and Peres (1995). (The
statement in that reference is slightly different, but the proof is the same.)
Lemma 11.3. Let ω be a general independent percolation on an infinite locally finite tree T .
Let W denote a Borel set of (infinite) rays in T starting at o, and suppose that P[ξ ⊂ ω] = 0
for each ξ ∈ W . Consider the random set ω∗ := {ξ ∈ W : ξ ⊂ ω}. Then a.s. ω∗ has no
isolated rays, so it is either empty or uncountable.
By an isolated ray in ω∗ we mean a ray whose intersection with the union of all other
rays in ω∗ is finite.
Proof of Theorem 11.1. For every vertex v ∈ V, let Xv be a network random walk starting
at v, independent from the other such walks. Let ω be the set of edges [v, v̂] such that X v
visits v̂. Then ω is an independent percolation with
£
¤
P [v, v̂] ∈ ω = h(v)/h(v̂) .
Take the WSF on T as generated by Wilson’s method rooted at infinity using the walks
Xv , according to some order hv0 , v1 , . . .i such that for every v 6= o, its parent v̂ appears
before it in the sequence. This defines a coupling of ω and the WSF. In this coupling, each
component of ω is contained in a component of the WSF, the WSF-component of o is the
union of the components of ω meeting the loop-erasure of Xo , and h(v) is the probability
that v is in the ω-component of o.
(a) By Theorem 11.2, the sum (11.1) diverges for all µ iff all components of ω are
finite a.s. In this case, all components in the WSF on T have one end a.s. Conversely, if
(11.1) converges for some µ ∈ M(∂T ), then ω has at least one infinite component a.s.; by
Lemma 11.3, the number of transient rays of such a component is 0 or ∞. Consequently,
the WSF on T a.s. has a component with more than one end. If h(v) → 0 as v → ∞,
then Lemma 11.3 implies that each infinite component of ω has uncountably many ends
a.s. Since every component of ω is contained in a component of the WSF, part (a) is
established.
§11. Analysis of the WSF on a Tree
49
(b) Using the notation of (a), it is easy to see that if all components of ω are recurrent
for the given resistances, then so are the components of the WSF; this depends on the
assumption that every infinite path in T is recurrent, and on the coupling. Moreover, it
clearly suffices to prove that the component of o in ω is recurrent a.s. By Theorem 11.2,
a subtree T ∗ ⊂ T containing o is recurrent for the resistances hR(e)i iff the intersection
of T ∗ with a certain independent percolation ω ′ on T has only finite components. In ω ′ ,
¡
¢−1
P
the probability that a vertex v is connected to o is pv := 1 + e R(e)
, where the
sum is over all edges on the path between v and o. The percolation ω ∩ ω ′ has no infinite
components a.s. iff
X
µ(v)
2
v6=0
µ
1
1
−
h(v)pv
h(v̂)pv̂
¶
=∞
for all µ ∈ M(∂T ) by Theorem 11.2. Since this sum dominates the sum
¶ X
µ
X
µ(v)2 R(v̂, v)
1
1
2
=
,
−
µ(v)
h(v)pv
h(v)pv̂
h(v)
(11.2)
v6=o
v6=0
it suffices to prove divergence of the latter for all µ ∈ M(∂T ). We then apply Fubini’s
theorem.
Write |w| := dist(o, w). If the path from o to v passes through w (i.e., v is a descendant
of w), write v > w. Write R(w ↔ ∞) for the effective resistance of the descendant subtree
of w, i.e., the minimal energy of a unit flow on this subtree with w as root. (See Lyons
and Peres (1997) for more background.) For any N ,
X µ(w)2 X µ(v)2
X µ(w)2
X µ(v)2 R(v̂, v)
>
R(v̂,
v)
>
R(w ↔ ∞) .
h(v)
h(w)
µ(w)2
h(w)
|v|>N
v>w
|w|=N
|w|=N
Denote the rightmost quantity by AN , and write
rw := R(ŵ, w) + R(w ↔ ∞) .
Using the identity
h(w) =
we obtain
AN +1 =
R(w ↔ ∞)
h(ŵ) ,
rw
X
|u|=N
1 X
µ(w)2 rw .
h(u)
ŵ=u
By the Cauchy-Schwarz inequality, for any vertex u,
X
µ(u)2
µ(w)2 rw > P
−1 .
r
w
ŵ=u
ŵ=u
§11. Analysis of the WSF on a Tree
50
Applying this to the preceding equality gives
AN +1 >
X
|u|=N
h(u)
µ(u)2
P
−1
ŵ=u rw
=
X µ(u)2
R(u ↔ ∞) = AN .
h(u)
|u|=N
Here we invoked the parallel-series laws for combining resistances in an electrical network
¡P
¢
−1 −1
to see that
= R(u ↔ ∞). Since A0 = R(o ↔ ∞) > 0, the tails of the series
ŵ=u rw
(11.2) are bounded away from 0, so the series diverges.
Given a tree T with root o and given k ∈ N, the level Tk is the set of vertices of T at
distance k from o. The tree is called spherically symmetric if for all k, every vertex in
Tk has the same number of children.
Corollary 11.4. (Spherically Symmetric Trees) Let T be a spherically symmetric
tree with levels hTk ik>0 . Suppose that for each k > 1, every edge connecting Tk−1 with Tk
P
is assigned resistance rk and the resulting network is transient, i.e.,
m rm /|Tm | < ∞.
P
Denote Ln := m>n rm /|Tm |. If
X
rn
= ∞,
(11.3)
2
|Tn | Ln Ln−1
n>1
then all components of the WSF on T have one end a.s.; if this series converges, then a.s.
all components of the WSF on T have uncountably many ends.
Note that the series in (11.3) converges if rn ≡ 1 and |Tn |/nγ is bounded above and below
by positive constants for some γ > 1.
Proof. For every vertex v ∈ Tn , we have h(v) = Pv [τo < ∞] = Ln /L0 . By convexity,
the sum in (11.1) is minimized by the µ ∈ M(∂T ) defined by µ(v) := 1/|Tn | for v ∈ Tn .
The first statement of the corollary now follows from Theorem 11.1(a). The existence of
a component with uncountably many ends when the series in (11.3) converges also follows
P
from Theorem 11.1(b) once we verify that in this case,
rn = ∞. To see this, note that
X
X Ln−1 − Ln
rn
∞>
=
|Tn |2 Ln Ln−1
|Tn |Ln Ln−1
n>1
n>1
¶ Xµ
¶
µ
X
1
1
1
1
>
−
−
=
|Tn |Ln
|Tn |Ln−1
|Tn |Ln
|Tn−1 |Ln−1
n>1
n>1
1
1
−
.
n→∞ |Tn |Ln
L0
= lim
Therefore, there is some c > 0 such that |Tn |Ln > c for all n. It follows that
X
X
X
rm >
crm /(|Tm |Lm ) >
crm /(|Tm |Ln ) = c
m>n
m>n
m>n
§12. Planar Graphs and Hyperbolic Lattices
for all n. This proves that
P
51
rm diverges.
It remains to show that when the series in (11.3) converges, then every component
has uncountably many ends a.s. Let ξ = hv0 , v1 , . . .i be an infinite ray in T starting
from o, and let ω be as in the proof of Theorem 11.1. Let qn be the probability that ω
contains an infinite ray starting at vn that does not contain any edges of ξ. Since there
are WSF-components with uncountably many ends, the Borel-Cantelli lemma implies that
P
n qn = ∞. Independence shows that ξ ∪ ω contains infinitely many infinite rays a.s.,
whence uncountably many by Lemma 11.3. This shows that To has uncountably many
ends a.s. A similar argument applies to the component of any vertex.
Remark 11.5. Let T be a tree. For the WSF on T corresponding to the conductances
C(e) := λ−dist(o,e) , with λ > 1 bounded above by the branching number of T , all component
trees have branching number at most λ a.s. (by Theorem 11.1 and Lyons (1990)). It is not
hard to see that equality holds if T has bounded degree, but not in general. When this
equality holds, the WSF interpolates continuously between the wired uniform spanning
forest (when λ = 1) and the whole tree (when λ is larger than the branching number of T ).
See Lyons (1990) for more information about random walks on trees and the branching
number.
§12. Planar Graphs and Hyperbolic Lattices.
A planar graph is a graph G embedded in the plane (in such a way that no two edges
cross each other). A planar network is a network (G, C), where G is a planar graph. A
face of a planar graph G is a component of R2 \ G. A planar network is proper if every
bounded set in the plane contains only finitely many edges and vertices.
Suppose that (G, C) is a finite or proper planar network. We define the dual network
(G , C † ) as follows. In each face f of G, we place a single vertex f † of G† . For every edge
†
e in G, we place an edge e† in G† connecting f1† and f2† , where f1 and f2 are the two faces
on either side of e. (It may happen that f1 = f2 ; then e† is a loop.) This is the usual
construction of the dual graph, G† . Note that G† is locally finite iff the boundary of
every face of G has finitely many vertices. Now set C † (e† ) := C(e)−1 for every e ∈ E.
When T is a set of edges of G, set
T ∗ := {e† : e ∈
/ T};
this is a subgraph of G† . The following is well known.
§12. Planar Graphs and Hyperbolic Lattices
52
Proposition 12.1. (Dual Trees) Let G be a finite connected planar network and G †
its dual. Let T be a spanning tree of G. Then T ∗ is a spanning tree of G† . Moreover,
weight(T )/ weight(T ∗ ) is independent of T .
Figure 2. A uniformly chosen wired spanning tree on a subgraph of Z2 , drawn by
Wilson (see Propp and Wilson (1998)).
Fig. 2 illustrates the situation. It reveals two spanning trees: one in white, the other
in black on the planar dual graph. Note that in the dual, the outer boundary of the grid
is identified to a single vertex.
Proof. T ∗ has no cycles because T is connected, and (V † , T ∗ ) is connected as T has no
cycles. It is easy to see that weight(T )/ weight(T ∗ ) is constant.
Theorem 12.2. (FSF is Dual to WSF) Let G be a proper planar network and G † its
dual. Suppose that G† is locally finite. Let T denote the FSF of G. Then T ∗ has the same
distribution as the WSF of G† .
Proof. Given a finite connected subgraph Gn of G with connected complement G \ Gn , let
G†n be its planar dual. Notice that G†n can be regarded as a finite subgraph of G† , but
with the outer boundary vertices identified to a single vertex. By Proposition 12.1, if T is
the weighted random spanning tree of Gn , then T ∗ is the weighted random spanning tree
of G†n . Thus, the theorem follows from the definitions of the FSF and the WSF.
Corollary 12.3. Let G be a proper planar network with G† locally finite. Then WSF =
FSF in G iff this happens in G† .
§12. Planar Graphs and Hyperbolic Lattices
53
Pemantle (1991) stated that the uniform spanning tree of Z2 has one end. We gave a
proof and extension in Theorem 10.3. An extension to graphs that need not be transitive
is:
Theorem 12.4. Let G be a proper planar network with G† locally finite and recurrent.
Then a.s. each component of the FSF of G has only one end.
Proof. Suppose that a component of FG , the FSF of G, has at least two ends with positive
probability. Then a bi-infinite path in it separates G† , which means that (V† , F∗G ) is
disconnected. By Theorem 12.2, it follows that the WSF of G† is disconnected with positive
probability, which is impossible on a recurrent graph by Proposition 5.6. We conclude that
each component of FG has only one end.
Similar reasoning shows:
Proposition 12.5. (Topology from Duality) Let G be a proper planar network with
G† locally finite. If each tree of the WSF of G has only one end a.s., then the FSF of G †
has only one tree a.s. If, in addition, the WSF of G has infinitely many trees a.s., then
the tree of the FSF of G† has infinitely many ends a.s.
On a Riemannian manifold M , a harmonic Dirichlet function f : M → R is a function
R
satisfying div ∇f = 0 and M |∇f |2 < ∞. There is an interesting phase transition between
dimensions 2 and 3 in hyperbolic space: HD(Hd ) ∼
= R for d = 1 and d > 3, but not for
d = 2. See Sario et al. (1977) or Dodziuk (1979).
Suppose that G is a graph of bounded vertex degree that is roughly isometric to a
manifold M with bounded local geometry. Kanai’s (1986) theorem says that M is transient
iff G is transient, while Holopainen and Soardi (1997) have shown that HD(G) ∼
= R iff
HD(M ) ∼
= R. Consequently:
Theorem 12.6. (Hyperbolic Phase Transition) Let G be a graph with bounded degrees that is roughly isometric to Hd . Then HD(G) ∼
= R iff d 6= 2.
A graph embedded in R2 or H2 is self-dual if it is isomorphic to its dual.
Taking stock, we arrive at the following surprising results:
Theorem 12.7. (WSF and FSF in Hd ) If G is a self-dual proper planar Cayley graph
roughly isometric to H2 , then the WSF of G has infinitely many trees a.s., each having
one end a.s., while the FSF of G has one tree a.s. with infinitely many ends a.s. If G is
a Cayley graph roughly isometric to Hd for some d > 3, then the WSF = FSF of G has
infinitely many trees a.s., each having one end a.s.
§12. Planar Graphs and Hyperbolic Lattices
54
Proof. In either case, each tree of the WSF has one end by Theorem 10.1. It follows from
Corollary 9.6 that a.s. the WSF has infinitely many trees. Now Theorems 12.6 and 7.3 and
Proposition 12.5 complete the proof.
An example of a self-dual Cayley graph roughly isometric to H 2 is shown in Fig. 3.
(See Chaboud and Kenyon (1996) for characterizations of planar Cayley graphs.)
Figure 3. A self-dual Cayley graph in the hyperbolic disc.
Remark 12.8. (Dropping Self-Duality) Actually, the assumption in the first part of
Theorem 12.7 that G is self-dual is not necessary. When G is not assumed to be self-dual,
the dual G† might not be transitive. However, one can show that the automorphism group
of G† is unimodular and its action on the vertices of G† (namely, the faces of G) has finitely
many orbits. One can, with some technical difficulties, generalize Theorem 10.1 to this
setting.
Here is a summary of the phase transitions.
Zd
d
Hd
2–4
>5
2
>3
FSF: trees
1
1
ends
1
∞
∞
WSF: trees
1
ends
1
1
∞
∞
∞
1
1
1
∞
1
Finally, we note a general corollary for transient planar graphs.
§13. The WSF in Nonamenable Graphs
55
Corollary 12.9. Let G be a transient planar network with bounded vertex conductance.
Then FSF 6= WSF on G and the WSF has infinitely many trees.
Proof. Benjamini and Schramm (1996a,b) proved that HD(G) 6∼
= R. Their results also
imply that two independent random walks in G intersect only finitely many times a.s., so
Theorem 9.4 applies.
An answer to Question 15.2 in Section 15 might provide a strengthening of this statement.
§13. The WSF in Nonamenable Graphs.
We have seen that the trees in the WSF of a Cayley graph have only one end. Here,
we consider their geometry. Let B(o, n) be a ball of radius n centered at a vertex o in
G. How much of the ball B(o, n) is taken up by the component To of the origin? For
Zd with d 6 4 this is, of course, the whole ball. For d > 5, the component retains the
“4-dimensionality” it has in Z4 : from the random walk estimates in Lawler (1991), it is
easily deduced that E|To ∩ B(o, n)|n−4 is bounded above and below by positive constants.
See Benjamini, Kesten, Peres and Schramm (1998) for more on this topic.
The spectral radius ρ(G) of a network G may be defined using the random walk
hX(n)i on G:
ρ(G) := lim sup(Px [X(n) = y])1/n .
n→∞
(The definition does not depend on x and y since G is connected.) The obvious inequality
Px [X(nk) = x] > Px [X(n) = x]k implies that Px [X(n) = x] 6 ρn for any x ∈ V and
n > 1. For the network with unit conductances on a finitely generated group G, Kesten
(1959) showed that ρ(G) = 1 iff G is amenable. More generally, for any network, ρ(G) < 1
is equivalent to a strong isoperimetric inequality (see Varopoulos (1985) or Gerl (1988)).
When ρ(G) < 1, the WSF-components are thinner than in Zd :
Theorem 13.1. (Tree Growth when ρ < 1) Let G be a graph with ρ(G) < 1 and
bounded vertex degree. Let o ∈ V be some basepoint. Denote by T o the component of o in
the WSF. Then c−1 n2 6 E|To ∩ B(o, n)| 6 c n2 for some 0 < c < ∞ and all n > 1.
Proof. We start with the upper bound. Let D be a bound on the vertex degrees. For any
vertex x and fixed k 6 m, we have
X
y∈G
Po [X(k) = y]Px [X(m − k) = y] 6 D
X
y∈G
Po [X(k) = y]Py [X(m − k) = x]
= DPo [X(m) = x] .
§13. The WSF in Nonamenable Graphs
56
Therefore, by Wilson’s method rooted at ∞,
P[x ∈ To ] 6
∞ X
m
XX
y∈G m=0 k=0
Po [X(k) = y]Px [X(m − k) = y] 6 D
∞
X
(m + 1)Po [X(m) = x] .
m=0
By the Cauchy-Schwarz inequality,
X
x∈B(o,n)
Consequently,
D
P
Po [X(m) = x] 6 |B(o, n)|
x∈B(o,n)
X
m6cn
2
X
Po [X(m) = x]2
x∈B(o,n)
6 D|B(o, n)|Po [X(2m) = o] .
P[x ∈ To ] is at most
(m + 1) + D 3/2 |B(o, n)|1/2
X
(m + 1)Po [X(2m) = o]1/2
(13.1)
m>cn
for every c. The hypothesis implies that Po [X(2m) = o] 6 ρ2m where ρ = ρ(G) < 1. Thus
by choosing c large enough, because |B(o, n)| 6 (D + 1)n , we can ensure that the second
summand in (13.1) tends to 0 as n → ∞, so that
E|To ∩ B(o, n)| =
X
x∈B(o,n)
P[x ∈ To ] 6
cD(n + 2)2
2
for all large n, which establishes the upper bound.
It remains to prove the lower bound on E|B(o, n) ∩ To |. For every v ∈ V, let Xv be a
simple random walk starting at v, with Xv , Xw independent when v 6= w. Let fu(o) denote
P
the loop erasure of Xo , as in Section 10, and denote by g(v, w) := k>0 P[Xv (k) = w] the
Green function for simple random walk on G. Observe that for any two vertices v, w at
distance k, we have
∞
D X
D
g(v, w) 6 P[w ∈ Xv ]g(w, w)Dg(w, v) 6
ρ2k ,
P[Xv (j) = v] 6
1−ρ
(1 − ρ)2
2
j=2k
so that g(v, w) 6 c0 ρk for some constant c0 .
The probability that a vertex v is in To is the probability that Xv intersects fu(o).
Now fu(o) contains vertices at every distance from o; we shall show that for some c 1 > 0
and any set S ⊂ B(o, n/2) that contains precisely one vertex at distance k from o whenever
0 6 k 6 n/2, we have
X
P[Lv (S) > 0] > c1 n2 ,
(13.2)
v∈B(o,n)
§13. The WSF in Nonamenable Graphs
where Lv (S) :=
P
w∈S
P
k>0
57
1{Xv (k)∈S} is the total occupation time of S by Xv .
Since every random walk starting at a vertex w ∈ B(o, n/2) must visit at least n/2
P
vertices before leaving B(o, n), we clearly have v∈B(o,n) g(w, v) > n/2, and therefore
X
E[Lv (S)] =
v∈B(o,n)
X
X
X
g(v, w) > D −1
v∈B(o,n) w∈S
X
g(w, v)
v∈B(o,n) w∈S
> D−1
X
n/2 > n2 /4D .
(13.3)
w∈S
By the Markov property,
E[Lv (S) | Lv (S) > 0] 6 max E[Ly (S)] .
y∈S
For any x, y ∈ S, we have dist(x, y) > |dist(o, x) − dist(o, y)|. Therefore,
E[Ly (S)] 6
X
2c0 ρk = c2
k>0
when y ∈ S, whence E[Lv (S)] 6 c2 P[Lv (S) > 0]. In conjunction with (13.3), this yields
(13.2).
By conditioning on fu(o), we obtain from (13.2) the bound
∀n E|B(o, n) ∩ To | > c1 n2 .
(13.4)
Remark 13.2. In the proof of the lower bound (13.4) given above, the hypothesis that
ρ(G) < 1 can be replaced by the weaker hypothesis that there is a summable decreasing
sequence hf (k)i, such that the Green function on G satisfies g(x, y) 6 f (dist(x, y)) for all
vertices x, y. This weaker hypothesis holds on any Cayley graph satisfying |B(o, n)| > cn 4
for all n; see Hebisch and Saloff-Coste (1993). In conjunction with Corollary 9.6, this
implies that (13.4) holds on any transient Cayley graph.
Remark 13.3. Classical results on critical branching processes imply that on a regular
tree (of degree at least 3), r(n) := n−2 |To ∩ B(o, n)| satisfies lim inf n→∞ r(n) = 0 and
lim supn→∞ r(n) = ∞. We omit the details.
We believe that the components of the WSF are recurrent on any graph (Conjecture 15.1). This was established for trees in Theorem 11.1 and can now be verified for
graphs with ρ < 1:
§13. The WSF in Nonamenable Graphs
58
Corollary 13.4. Let G be a graph with ρ(G) < 1 and bounded vertex degree. Let o ∈ V
be some basepoint. Then a.s. the WSF-component To is recurrent.
The corollary follows immediately from Theorem 13.1 and the following general lemma,
since the distance from a vertex v to o in To is bounded below by the distance in G.
Lemma 13.5. Let Υ be a random graph with a distinguished vertex o, and denote by Υ j the
Pn
set of edges with an endpoint at distance j from o. If E j=1 |Υj | 6 cn2 for some c < ∞
and all n > 1, then simple random walk on Υ is recurrent a.s.
P
P
Proof. Let han i be a decreasing positive sequence such that n an = ∞ and n na2n < ∞,
e.g., an = (n log(n + 1))−1 . By the Cauchy-Schwarz inequality,
∞=
Ã
∞
X
an
n=1
!2
6
Ã
∞
X
1
|Υn |
n=1
!Ã
∞
X
n=1
!
a2n |Υn |
.
(13.5)
However,
E
Ã
∞
X
n=1
!
a2n |Υn |
=
∞
X
n=1
6
∞
X
n=1
Hence, (13.5) gives
P∞
n=1
n
X
|Υj |
(a2n − a2n+1 )E
j=1
(a2n
−
a2n+1 )cn2
=
∞
X
n=1
a2n c(2n − 1) < ∞ .
1/|Υn | = ∞ a.s. By the Nash-Williams criterion (see Doyle and
Snell (1984)), this implies that Υ is recurrent a.s.
Pn
Remark 13.6. The hypothesis of Lemma 13.5 can be replaced by E j=1 |Υj | 6 nbn ,
P −1
where hbn i is an increasing sequence such that
bn = ∞ and supn n(bn − bn−1 )/bn < ∞.
Pn
P
−1
To verify this, define an := (bn Dn ) where Dn := k=1 b−1
an = ∞ but
k , observe that
P 2
an bn < ∞, and mimic the proof above.
In Zd , d > 4, the WSF is not connected. How close is it to being connected? One
variant of this question was addressed in Remark 9.7.
Here is another variant. Suppose that ǫ ∈ (0, 1). Let ωǫ be Bernoulli(ǫ) percolation;
that is, for all e ∈ E, let e ∈ ωǫ with probability ǫ, independently for different e’s, and
independent of the WSF, F. Is F ∪ ωǫ connected?
Burton and Keane (1989) have shown that a random subgraph of Z d whose distribution
is invariant under translations and satisfies the so-called “finite-energy” condition has a.s.
at most one infinite component. Although F ∪ ωǫ satisfies only half of that condition,
namely, that edges can be added without great penalty, the Burton-Keane argument does
§13. The WSF in Nonamenable Graphs
59
show that F ∪ ωǫ and also ωǫ have a.s. at most one infinite component in any transitive
amenable network. Since ǫ can be taken arbitrarily small and since F \ ω ǫ has no infinite
components a.s., one can argue that this demonstrates that uniform spanning forest on Z d
is a critical model: it exhibits criticality with respect to connectivity.
In contrast, it has been conjectured by Benjamini and Schramm (1996c) that on any
nonamenable transitive graph, there is some ǫ such that ωǫ has infinitely many infinite
components. Although this conjecture is still unresolved, the following theorem shows
that for any nonamenable network, F ∪ ωǫ has a.s. infinitely many infinite components if
ǫ > 0 is sufficiently small.
Theorem 13.7. (WSF + ǫ) Let G be a network with edge conductances bounded below and
above and bounded vertex-degrees. Assume that the spectral radius ρ of G is less than 1.
Let F be the WSF of G. For any ǫ ∈ (0, 1), let ωǫ be Bernoulli(ǫ) percolation on G. Then,
provided ǫ is sufficiently close to 0, a.s. there are infinitely many components in F ∪ ω ǫ and
each has infinitely many ends.
We first show:
Lemma 13.8. Let G be a network where the probability that two random walk paths starting
at v and w will intersect tends to 0 as dist(v, w) → ∞. Then a.s. each component of the
WSF has an infinite boundary as a subgraph of G.
Proof. Consider a finite (possibly empty) vertex set A. We need to show that the probability that A is the boundary of a component of the WSF is zero. Let W be any infinite
component of Ac . By applying the hypothesis to two sufficiently far apart vertices v, w in
W , we see that the probability that W is in one component of the WSF is arbitrarily small,
hence 0. Thus A is a.s. not the boundary of a WSF tree. Since there are only countably
many finite sets of vertices, this completes the proof.
Proof of Theorem 13.7. Adding loops if necessary, we may assume for convenience that all
the vertex conductances are equal. (This may change ρ, but will not make ρ(G) equal 1.)
It then follows that for every v, u ∈ V and all n > 0,
Pv [X(n) = u] 6 Pv [X(2n) = v]1/2 6 ρn ,
(13.6)
where X is the random walk on G.
Given o, v ∈ V, let N (o, v) be the number of distinct simple paths from o to v in F∪ω ǫ .
We shall show that for all ǫ > 0 sufficiently small,
∀o, v ∈ V
E[N (o, v)] < ∞
(13.7)
§13. The WSF in Nonamenable Graphs
60
and
lim
dist(o,v)→∞
E[N (o, v)] = 0 .
(13.8)
Let us first see that this will suffice to prove the theorem. Note that E[N (o, v)] bounds
the probability that o and v are in the same F∪ωǫ -component. Consequently, (13.8) implies
that a.s. F ∪ ωǫ has infinitely many components. There is an obvious map ψ from the set
of ends of components of F to the set of ends of components of F ∪ ωǫ . Suppose that v
and o belong to distinct components of F, but to the same component of F ∪ ω ǫ . Let ξv
be an end of the component of F containing v and let ξo be an end of the component of
F containing o. If ψ(ξo ) = ψ(ξv ), then it easily follows that N (o, v) = ∞. Consequently,
ψ is a.s. injective when (13.7) holds. By the preceding lemma, each component of F has
infinite boundary a.s. Therefore, a.s. each component of F has infinitely many edges in ω ǫ
connecting it to other components of F. For every pair T1 , T2 of components of F, there
are only finitely many edges in ωǫ joining them, by injectivity of ψ. Consequently, every
component T∗ of F ∪ ωǫ contains infinitely many components of F; therefore, T∗ must have
infinitely many ends, by another application of injectivity of ψ. Consequently, it is enough
to prove (13.7) and (13.8).
We now think of F as oriented towards infinity, that is, we consider the OWSF. Recall
that at every vertex w of F, there is precisely one outgoing edge of F.
Consider a basepoint o ∈ V and some z ∈ V \ {o}. Suppose that P is a simple path
from o to z in F ∪ ωǫ . Then there is a unique sequence γ(F, ωǫ , P) of the form
(v1 , w1 , u1 , v2 , w2 , u2 , . . . , un )
(13.9)
with the following properties:
(a) v1 = o, un = z;
(b) for each j = 1, . . . , n, there is a path Pj+ in F ∩ P from vj to wj , and the
orientation of this path agrees with the orientation of F and of P;
(c) for each j = 1, . . . , n, there is a path Pj− in F ∩ P from wj to uj that agrees
with the orientation of P and goes opposite to the orientation on F;
(d) all the paths Pj± are pairwise vertex disjoint, except that Pj+ and Pj− share
the vertex wj ; and
(e) for each j = 1, . . . , n − 1, there is an edge in ωǫ connecting uj and vj+1 , but
there is no such edge in F.
Note that we may have vj = wj or wj = uj ; that is, some of the paths Pj± may have
only one vertex. To obtain this sequence γ(F, ωǫ , P), just follow P and record every vertex
where the orientation changes or where an edge of ωǫ \ F is used. Note that given F, ωǫ ,
§13. The WSF in Nonamenable Graphs
61
and β of the form (13.9), there is at most one simple path P from v1 to un in F ∪ ωǫ such
that β = γ(F, ωǫ , P).
We are going to compare the probability of finding a sequence γ(F, ω ǫ , P) in F ∪ ωǫ
to the probability of finding it in the image of the network random walk. Let q(β) be the
probability that there is some simple path P in F∪ωǫ from o to z such that β = γ(F, ωǫ , P).
Say that a sequence β of the form (13.9) is adapted to a finite or infinite path
y(0), y(1), . . . in G if y(0) = v1 and there are integers 0 6 t1 6 t′1 < t2 6 t′2 < · · · 6 t′n such
that y(tj ) = wj and y(t′j ) = uj for j = 1, . . . , n and y(t′j + 1) = vj+1 for j = 1, . . . , n − 1.
Let qX (β) be the probability that β is adapted to X, where X(0), X(1), . . . denotes the
network random walk that starts at X(0) = o.
Lemma 13.9. There is a constant c > 0, depending only on the network G, such that for
all β of the form (13.9) such that each uj neighbors in G with vj+1 (j = 1, . . . , n − 1),
q(β) 6 (cǫ)n−1 qX (β) .
Proof. Construct F by Wilson’s method rooted at infinity, starting with the vertices
w1 , w2 , . . . , wn , v1 , v2 , . . . , vn , u1 , u2 , . . . , un
in this order. For β to occur as γ(F, ωǫ , P) for a simple path P ⊂ F ∪ ωǫ from v1 to un , the
random walk starting at each vj and each uj must hit the corresponding wj , and [uj , vj+1 ]
must be in ωǫ for each appropriate j. Let ϕ(v, w) denote the probability that a network
random walk that starts at v will hit w. Then we get
q(β) 6 ǫ
n−1
n
Y
¡
¢
ϕ(vj , wj )ϕ(uj , wj ) .
j=1
Reversibility and the equality of conductances at vertices imply that ϕ(v, w) = ϕ(w, v).
Since uj neighbors with vj+1 when j = 1, . . . , n − 1, the transition probabilities satisfy
p(uj , vj+1 ) > 1/c for some constant c > 0 depending only on G. Consequently,
q(β) 6 ǫn−1
n ³
´
Y
ϕ(vj , wj )ϕ(wj , uj )
j=1
6 (cǫ)
n−1
6 (cǫ)
n−1
n
Y
¡
ϕ(vj , wj )ϕ(wj , uj )
j=1
which proves the lemma.
qX (β) ,
Y
¢ n−1
j=1
p(uj , vj+1 )
§13. The WSF in Nonamenable Graphs
62
We now continue with the proof of Theorem 13.7. Let Ym be the set of all walks
y = hy(0), . . . , y(m)i with y(0) = o, y(j) ∼ y(j − 1) for j = 1, . . . , m, and y(m) = z. Let
Bn be the set of all sequences β of the form (13.9) with v1 = o, un = z, uj ∼ vj+1 , all
uj distinct, all vj are distinct, and all wj distinct. Given y ∈ Ym , let Bn (y) be the set of
β ∈ Bn adapted to y. Note that
∀y ∈ Ym
Therefore, by Lemma 13.9, we have
E[N (o, z)] =
6
∞
X
X
n=1 β∈Bn
∞
X
X
¯
¯
¯Bn (y)¯ 6
µ ¶3 µ ¶
m
3m
6
.
n
3n
q(β)
(cǫ)n−1 qX (β)
n=1 β∈Bn
6
∞
X
X
m/3
Po [∀j = 1, . . . , m X(j) = y(j)]
n=1
m=1 y∈Ym
6
∞
X
m/3
Po [X(m) = z]
6 (cǫ)−1
X
(cǫ)
n−1
n=1
m=1
∞
X
m=1
X
µ
3m
3n
¶
¯
¯
(cǫ)n−1 ¯Bn (y)¯
³
´3m
.
Po [X(m) = z] 1 + (cǫ)1/3
With (13.6), this gives
E[N (o, z)] 6 (cǫ)−1
X
m>dist(o,z)
´3m
³
1/3
.
ρm 1 + (cǫ)
This implies (13.7) and (13.8), which complete the proof.
We call a random subgraph of G an automorphism-invariant percolation if its
distribution is invariant under the automorphism group of G. The Burton-Keane (1989)
argument has two parts; one shows that all automorphism-invariant percolations on an
amenable transitive graph have a.s. at most 2 ends in each component. Theorem 13.7
provides a converse, which can be sharpened as follows:
Corollary 13.10. (Burton-Keane Converse) Let G be a nonamenable connected
graph with a unimodular transitive automorphism group. Then there is an automorphisminvariant percolation process ψ on G in which each component is a tree with infinitely many
ends.
§14. Applications to Loop-Erased Walks and Harmonic Measure
63
Proof. By Lemma 7.4 of BLPS (1999), any invariant percolation (in our case, F ∪ ω ǫ ) with
more than two ends in some component can be thinned out to an invariant forest ψ ′ with
more than two ends in some component. By Theorem 7.2 of that paper, some component
of ψ ′ will have pc < 1 with positive probability, hence infinitely many ends. Condition
on that event, and let ψ ′′ be ψ ′ with all trees that have finitely many ends removed. For
each vertex v ∈ V that is not in an infinite component of ψ ′′ , choose randomly, uniformly
and independently an edge e = e(v) that connects it to a vertex closer to the infinite
components of ψ ′′ . Let ψ be the union of ψ ′′ and all such edges e(v). Then ψ satisfies the
requirements.
In Benjamini, Lyons and Schramm (1999), Theorem 13.7 is used to show that under
the same assumptions as in Corollary 13.10, there is an automorphism-invariant random
forest F ⊂ G with ρ(F) < 1 a.s.
We can extend Proposition 10.11 to the planar non-transitive setting:
Corollary 13.11. Let G be a proper planar graph with bounded degrees and a bounded
number of sides to its faces. If ρ(G) < 1, then a.s. some component T of the FSF on G
has pc (T ) < 1.
Proof. Let F be the FSF of G, and let ωǫ be a Bernoulli(ǫ) percolation independent of
F. Recall from Theorem 12.2 that F∗ := {e† : e ∈
/ F} has the same distribution as the
WSF on G† . Consequently, F \ ωǫ has the same distribution as F∗ ∪ {e† : e ∈ ωǫ }. Since
{e† : e ∈ ωǫ } is Bernoulli(ǫ) percolation on G† , the result follows from Theorem 13.7,
because F \ ωǫ has an infinite component whenever F∗ ∪ {e† : e ∈ ωǫ } has more than one
infinite component.
§14. Applications to Loop-Erased Walks and Harmonic Measure.
Infinite loop-erased random walk is defined in any transient network by chronologically
erasing cycles from the random walk path. On a recurrent network, the natural substitute
is to run random walk until it first reaches distance n from its starting point, erase cycles,
and take a weak limit as n → ∞. On a general recurrent network, such a weak limit
need not exist; in Z2 , weak convergence was established by Lawler (1988) using Harnack
inequalities (see Lawler (1991), Prop. 7.4.2). Lawler’s approach yields explicit estimates of
the rate of convergence, but is difficult to extend to other networks. Using spanning trees,
we obtain the following general result.
§14. Applications to Loop-Erased Walks and Harmonic Measure
64
Proposition 14.1. Let hGn i be an exhaustion of a recurrent network G. Consider the
network random walk hXo (k)i started from o ∈ G. Denote by τGcn the first exit time of Gn ,
and let Ln be the loop erasure of the path hXo (k) : 0 6 k 6 τGcn i. If the random spanning
tree TG in G has one end a.s., then the random paths Ln converge weakly to the law of the
unique ray from o in TG . In particular, this applies if G is a proper planar network with
a locally finite recurrent dual.
Proof. This is immediate from the definition of the WSF, Wilson’s method applied to the
wired graph GW
n , and Proposition 5.6. The final assertion uses Theorem 12.4.
Let A be a finite set of vertices in a recurrent network G. Denote by τA the hitting
time of A, and by hA
v the harmonic measure from v on A:
∀B ⊆ A hA
v (B) := Pv [Xv (τA ) ∈ B] .
If the measures hA
v converge when dist(v, A) → ∞, then it is natural to refer to the limit
as harmonic measure from ∞ on A. This convergence fails in some recurrent networks
(e.g., in Z), but it does hold in Z2 ; see Lawler (1991, Thm. 2.1.3). As above, random
spanning trees yield a very general result.
Theorem 14.2. Let G be a recurrent network and A be a finite set of vertices. Suppose
that the random spanning tree TG in G has one end a.s. Then the harmonic measures hA
v
converge as dist(v, A) → ∞.
Proof. Add a finite set B of edges to G to form a graph G′ in which the subgraph (A, B)
is connected, and suppose that B is minimal with respect to this property. Assign unit
conductance to the edges of B. Note that having at most one end in T is a tail event.
Since TG′ conditioned on TG′ ∩ B = ∅ has the same distribution as TG , by tail triviality,
a.s. TG′ has one end. Similarly, because TG′ conditioned on TG′ ∩ B = B has the same
distribution as TG′ /B , also TG′ /B has one end a.s.
The path from v to A in TG′ /B is constructed by running a random walk from v until
it hits A and then loop erasing. Thus, when dist(v, A) → ∞, the measures h A
v must tend
to the conditional distribution, given TG′ ∩ B = B, of the point in A that is closest (in
TG′ /B ) to the unique end of TG′ /B .
If the FSF in a network G is a tree T , then for any two adjacent vertices v, w ∈ G,
there is a unique simple path P(v, w) in T that connects them; when G is recurrent,
Proposition 5.6 shows that this path can be obtained by loop erasing the network random
walk started at v and stopped at w. An easy lower bound for the tail probabilities of the
random variable diam P(v, w) is given in the following theorem. It would be interesting to
obtain precise estimates for these probabilities; see Lawler (1999) for a recent improvement.
§15. Open Questions
65
Theorem 14.3. (Connection Diameter Tail) Consider the uniform spanning forest
in Zd , 2 6 d 6 4, and let v, w be adjacent vertices in Zd . Then
∀n > 1
P[diam P(v, w) > n] >
1
.
8n
(14.1)
Proof. We use the coordinates (x1 , . . . , xd ) in Zd . Let ∂S be the perimeter of the square
S := [−n, n]2 × {0}d−2 ; we think of ∂S both as a closed path and as a set of 8n edges. For
each edge e = [v, w] in ∂S, we may consider P(e) = P(v, w) as a detour for e. If for all
e in ∂S, the path P(e) had diameter less than n, then the concatenation of these paths
would be a closed path homotopic to ∂S in Rd − {x : x1 = x2 = 0}. Consequently, T
would contain a cycle, which is a contradiction. Thus
X
1{diam P(e)>n} > 1 .
e∈∂S
Taking expectations and using the edge-transitivity of Zd proves (14.1).
§15. Open Questions.
There are many tantalizing open questions related to uniform spanning forests. We
present a sample here.
Conjecture 15.1. Each component tree of the wired uniform spanning forest on any graph
G is recurrent a.s. for simple random walk. More generally, if the edges in the components
of the WSF on a network with bounded conductance are given the conductances they have
in the network, then all the components are recurrent a.s.
This holds when G is a transitive network with Aut(G) unimodular, when G is a
tree, and when G is a graph satisfying ρ(G) < 1, by Lemma 10.2, Theorem 11.1 and
Corollary 13.4, respectively.
Remark (added February 2000): This conjecture has just been proved by Benjamin
Morris (personal communication).
Question 15.2. Let G be a proper transient planar graph with bounded degree and a
bounded number of sides to its faces. Is the free spanning forest a single tree a.s.? If true,
this would strengthen Corollary 12.9.
Question 15.3. If a graph has spectral radius < 1, must each tree in its wired uniform
spanning forest have one end a.s.? We know that each tree is recurrent a.s., by Corollary 13.4.
§15. Open Questions
66
Question 15.4. Does each component in the wired uniform spanning forest on an infinite
supercritical Galton-Watson tree have one end a.s.?
Question 15.5. Let G be a transitive network whose automorphism group is not unimodular. Does every tree of the WSF on G have one end a.s.?
The following question was suggested to us by O. Häggström:
Question 15.6. Let G be a transitive network. By Remark 9.8, the number of trees of
the FSF is a.s. constant. Is it 1 or ∞ a.s.?
Question 15.7. Let G be an infinite network. Is the number of trees of the FSF a.s.
constant?
Question 15.8. Let G be a transitive network with WSF 6= FSF. Must all components of
the FSF have infinitely many ends a.s.?
In view of Proposition 10.11, this would follow in the unimodular case from a proof
of the following conjecture:
Conjecture 15.9. The components of the FSF on a unimodular transitive graph are indistinguishable in the sense that for every automorphism-invariant property A of subgraphs,
either a.s. all components satisfy A or a.s. they all do not. The same holds for the WSF.
This fails in the nonunimodular setting, as the example in Lyons and Schramm (1999)
shows.
Question 15.10. Is there a “natural” monotone coupling of FSF and WSF? For example,
if G is a Cayley graph, is there a monotone coupling that is invariant under multiplication
by elements of G?
Question 15.11. Let G be an infinite network such that WSF 6= FSF on G. Does it follow
that WSF and FSF are mutually singular measures?
This question has a positive answer for trees (there is exactly one component FSF-a.s.
on a tree, while the number of components is a constant WSF-a.s. by Theorem 9.4) and
for networks G where Aut(G) has an infinite orbit (Corollary 8.2).
Conjecture 15.12. Let To be the component of the identity o in the WSF on a Cayley
graph, and let ξ = hvn : n > 0i be the unique ray from o in To . The sequence of “bushes”
hbn i observed along ξ converges in distribution. (Formally, bn is the connected component
of vn in T \ vn−1 \ vn+1 , multiplied on the left by vn−1 .)
Question 15.13. (This question was also asked by J. Propp; see Propp (1997).) One may
consider the uniform spanning tree on Z2 embedded in R2 . In fact, consider it on ǫZ2 in
§15. Open Questions
67
R2 and let ǫ → 0. In what sense should a limit be taken, and how can one show the limit
exists? Does the limit have some conformal invariance property?
Some reasonable answers to the question of how to define the scaling limit have been
given (following the circulation of an earlier draft of this paper) by Aizenman, Burchard,
Newman, and Wilson (1999) and Schramm (2000). Still, the existence and conformal
invariance of the limit remain open.
The invariance asked for in Question 15.13 could be expected on the basis of conformal
invariance of simple random walk. It is also supported by some computer simulations of
O. Schramm. In addition, the uniform spanning tree is intimately tied to random domino
tiling of Z2 : see, e.g., Burton and Pemantle (1993). Kenyon (1997) shows that domino
tilings have a strong form of conformal invariance and proves the conformal invariance
of certain properties in Kenyon (2000). Nonrigorous conformal field theory was used by
Duplantier (1992) and Majumdar (1992) to estimate the rate of escape of loop-erased walks
in Z2 . Finally, critical Bernoulli percolation is believed to have conformal invariance in the
limit (see, e.g., Langlands, Pouliot and Saint-Aubin (1994) and Benjamini and Schramm
(1998)) and the uniform spanning tree is a critical model, as explained in Section 13.
Figure 4. The inside of this lattice-filling curve is a uniform spanning tree on a square
grid, while the outside is another on the dual grid (wired).
References
68
Tóth and Werner (1998), §11, explain the connection between certain self-repelling
random walks on Z and simple oriented random walk on Z2 that is oriented so that only
steps to the right and up are possible. Of course, simple oriented random walk never visits
any state more than once. Tóth and Werner consider coalescing paths of this Markov chain;
these paths form the wired random spanning tree of Z2 that is associated to this chain by
Wilson’s method rooted at infinity as in Remark 5.4. The dual of this tree is considered,
as well as the lattice-filling curve that “threads” between the two trees; see Fig. 4 for
the lattice-filling curve of a uniform spanning tree. Their paper is devoted to analyzing
the continuous analogue of these objects in the oriented case. In particular, a stochastic
differential equation describes their space-filling curve. Note that the lattice-filling curve
completely determines the tree and its dual. There is also a stochastic differential equation
that is conjectured to determine the space-filling curve that presumably is the limit as ǫ → 0
of the lattice-filling curves threading between the uniform spanning tree and its dual on
ǫZ2 . More details on this conjecture appear in Schramm (2000).
Acknowledgement. We thank Dayue Chen, Olle Häggström, Ben Morris, Elchanan Mossel, Jim Propp and Jeff Steif for remarks on a previous version and Greg Lawler for references.
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Mathematics Department, The Weizmann Institute of Science, Rehovot 76100, Israel
itai@wisdom.weizmann.ac.il
http://www.wisdom.weizmann.ac.il/~itai/
Department of Mathematics, Indiana University, Bloomington, IN 47405-5701, USA
rdlyons@indiana.edu
http://php.indiana.edu/~rdlyons/
Institute of Mathematics, The Hebrew University, Givat Ram, Jerusalem 91904, Israel and
Department of Statistics, University of California, Berkeley, CA 94720-3860, USA
peres@stat.berkeley.edu
http://www.stat.berkeley.edu/~peres/
Mathematics Department, The Weizmann Institute of Science, Rehovot 76100, Israel
schramm@wisdom.weizmann.ac.il
http://www.wisdom.weizmann.ac.il/~schramm/