Bounding a Random Environment for Two-dimensional Edge-reinforced Random Walk

We consider edge-reinforced random walk on the inﬁnite two-dimensional lattice. The process has the same distribution as a random walk in a certain strongly dependent random environment, which can be described by random weights on the edges. In this paper, we show some decay properties of these random weights. Using these estimates, we derive bounds for some hitting probabilities of the edge-reinforced random walk


Introduction
Definition of the model.Linearly edge-reinforced random walk (ERRW) on Z 2 is the following model: Consider the two-dimensional integer lattice Z 2 as a graph with edge set (1.1) Here, | • | denotes the Euclidean norm.In particular, the edges are undirected.Fix a vertex v 0 ∈ Z 2 and a positive number a > 0. A non-Markovian random walker starts in X 0 = v 0 .At every discrete time t ∈ N 0 , it jumps from its current position X t to a neighboring vertex X t+1 in Z 2 , |X t+1 − X t | = 1.The law P v 0 ,a of the random walker is defined in terms of the time-dependent weights 1 {e={Xs,X s+1 }} , e ∈ E. (1.2) The weight w e (t) of edge e at time t equals the number of traversals of e up to time t plus the initial weight a.Thus, w e (t) increases linearly in the number of crossings of e.The transition probability P v 0 ,a [{X t , X t+1 } = e | X 0 , . . ., X t ] is proportional to w e (t) for all edges e ∋ X t : e ′ ∋Xt w e ′ (t) This model was introduced by Diaconis [Dia88].Throughout this paper, "edge-reinforced random walk" will always mean "linearly edge-reinforced random walk", as defined above.We do not consider any other reinforcement scheme.
Main open problem and previous results for Z d .As Pemantle [Pem88] remarks, Diaconis asked whether this process on Z 2 and more generally on Z d , d ≥ 2, is recurrent.This question is still open almost twenty years after it was raised.Even worse, proving anything about edgereinforced random walk on Z 2 tends to be hard.To our best knowledge, almost nothing is known about edge-reinforced random walk on Z d for any d ≥ 2, with only a few exceptions: 1. Sellke [Sel94] has proved for edge-reinforced random walk on Z d in any dimension d, that almost surely the range is infinite, and that the random walker hits each coordinate plane x k = 0, k = 1, . . ., d almost surely infinitely often.
2. Recently, in [MR07c], we have shown that the edge-reinforced random walk on any locally finite graph has the same distribution as a random walk in a random environment given by random, time-independent, strictly positive weights (x e ) e on the edges.As a consequence, the edge-reinforced random walk on any infinite, locally finite, connected graph visits all vertices infinitely often with probability one if and only if it returns to its starting point almost surely.
Up to now, except for the present paper, there seem to be no further rigorous results about edge-reinforced random walk on Z 2 .All attempts to apply coupling methods or monotonicity arguments to this problem have failed so far.Monte Carlo simulations seem to indicate nonrigorously that -at least for some initial weights -edge-reinforced random walk on Z 2 typically does not stay very close to its starting point.Rather, the random walker spends most of its time at some, possibly infinitely many, small islands located at varying distance from the starting point.This indicates that the weights (x e ) e in (b) might vary in rather small regions over many orders of magnitude.
Intention of this paper.In this paper, we show for an edge-reinforced random walk on Z 2 that at least in a weak, stochastic sense, the weights x e in (b) converge to 0 as e gets infinitely far from the origin.This provides the first analysis of the global behaviour of the random environment over Z 2 .Bounds on the weights x e induce information for the law of the edgereinforced random walk.For example, our bounds allow for the first time the estimation of hitting probabilities of the paths of edge-reinforced random walk on Z 2 .
The present paper is independent of our papers [MR07a], [MR05], [Rol06] on edge-reinforced random walk on ladders, as the transfer operator technique used there breaks down in two dimensions.
Previous results on other graphs.For graphs where the edge-reinforced random walk is recurrent, a representation different from the one in [MR07c] follows from the paper [DF80] by Diaconis and Freedman, which was written before edge-reinforced random walk was introduced.
For finite graphs, Coppersmith and Diaconis [CD86] discovered an explicit, but complicated formula for the joint law of the fraction of time spent on the edges.In fact, this joint law just equals the law of the weights (x e ) e normalized appropriately; see [Rol03].A proof of the formula describing the joint law of (x e ) e is published in [KR00].The random environment is strongly dependent, unless the graph is a tree-graph.
Pemantle [Pem88] examined the model on infinite trees; in particular, he showed that the process on tree graphs has the same distribution as a random walk in an independent random environment.
On infinite ladders, for sufficiently large initial weights a, the random environment can be described as an infinite-volume Gibbs measure.It arises as an infinite-volume limit of finite-volume Gibbs measures; see [MR05], [Rol06], [MR07a].The finite-volume Gibbs measures are just a reinterpretation of the formula of Coppersmith and Diaconis in the version described by Keane and Rolles.This reinterpretation has been used in the above references to prove recurrence of the process on ladders and also to analyze the asymptotic behavior of the edge-reinforced random walk in more detail.
In [MR07b], we have recently proven recurrence of the edge-reinforced random walk on some fully two-dimensional graphs different from Z 2 .In the present paper, we push the method used for these graphs to its limits: Unlike in [MR07b], in the present paper, we prove bounds for the random weights in infinite volume, and we prove strong bounds for the expected logarithm of the random weights.These bounds require more sophisticated deformations than in [MR07b].However, the present paper can be read independently of [MR07b].
More information about the history of linearly edge-reinforced random walk can be found in [MR06b].

Results
We present results for edge-reinforced random walk on Z 2 ("infinite volume"), but also on finite boxes in Z 2 with periodic boundary conditions ("finite volume").The latter results are used in the proof of the former ones.

Hitting probabilities for ERRW
For v ∈ Z 2 , let τ v := inf{n ≥ 1 : X n = v} denote the first time ≥ 1 when the random walker visits v.
Theorem 2.1 (Hitting probabilities for ERRW).For all a > 0, there are c 1 (a) > 0 and β(a) > 0, such that for all v ∈ Z 2 \ {0}, the following hold: 1.The probability to reach the vertex v before returning to the vertex 0 is bounded by 2. For all n ≥ 1, the probability that the random walker visits the vertex v at time n satisfies the same bound P Although these estimates are weak if compared with what is known for simple random walk on Z 2 , to our knowledge, no bounds of any similar type were known before.

Bounds in infinite volume
Set Ω = (0, ∞) E .For x ∈ Ω, let Q v 0 ,x denote the law of a Markovian nearest neighbor random walk on (Z 2 , E) with starting point v 0 in a time-independent environment given by the edge weights x.Given this random walk is at u and given the past of the random walk, the probability it jumps to the neighboring vertex v is proportional to the weight x e of the edge e = {u, v}.
Edge-reinforced random walk on Z 2 can be represented as a random walk in a random environment (RWRE).More precisely: Theorem 2.2 (ERRW as RWRE, Theorem 2.2 in [MR07c]).Let a > 0. There is a probability measure Q 0,a on Ω, such that for all events A ⊆ (Z 2 ) N 0 , one has It is not known whether Q 0,a is unique up to normalization of the edge weights x.In this article, we prove some decay properties of the random environment.These results hold for at least one choice of Q 0,a .For x ∈ Ω and v ∈ Z 2 , set (2.8) The following two theorems show that in some weak probabilistic sense, the ratios of weights x v /x 0 tend to zero as |v| → ∞.We phrase two formal versions of this statement.The first version only cares about the expected logarithms of ratios x v /x 0 .In this version, we get a fast divergence to −∞ as |v| → ∞ if only a > 0 is small enough.
Theorem 2.3 (Decay of the expected log weights).There exist functions c 2 , c 3 : (0, ∞) → (0, ∞) with c 2 (a) → ∞ as a ↓ 0, such that the following holds: For all a > 0 and all v ∈ Z 2 \ {0}, (2.9)Such a fast decay of the logarithms of the weights in a stochastic sense has not been proven before on any fully two-dimensional graph.The methods used in [MR07b] do not suffice to prove such a bound.
However, Theorem 2.3 does not imply weak convergence of x v /x 0 to zero.Weak convergence is stated among others in the following theorem, but our bound for the rate of convergence is not as strong as in the preceding theorem.
The estimates for the hitting probabilities of the edge-reinforced random walk in Theorem 2.1 are derived from the bounds in the preceding theorem.
Let e, f ∈ E be two neighboring edges, i.e. two edges containing a common vertex v.The random variables log(x e /x f ) with respect to Q 0,a are tight with exponential tails, uniformly in the choice of the edges e and f .This is stated formally in the following theorem: Theorem 2.5 (Tightness of ratios of neighboring edge weights).For all a > 0 and all α ∈ (0, a/2), one has (2.11)

Uniform bounds in finite volume
All our infinite-volume results are derived from uniform finite-volume analogs for edge-reinforced random walk on finite boxes."Uniform" here means "uniform in the size of the finite box".
We consider a (2N + 1) × (2N + 1) box ) .If there is no risk of confusion, we identify V (N ) with the subset Ṽ (2.13) For an equivalence class v (N ) = v + (2N + 1)Z 2 , set This is just the Euclidean distance of v (N ) to the origin, viewed as an element of Ṽ (N ) .
Just as in the infinite-volume case, for v 0 ∈ V (N ) and a > 0, let P (N ) v 0 ,a denote the law of edgereinforced random walk on (V (N ) , E (N ) ) with starting point v 0 and constant initial weights a. Set Ω (N ) = (0, ∞) E (N ) .For x ∈ Ω (N ) , let Q (N ) v 0 ,x denote the law of a random walk on (V (N ) , E (N ) ) with starting point v 0 in a time-independent environment given by weights x.Note that multiplying all components of x by the same (possibly x-dependent) scaling factor α does not change the law of the corresponding random walk.The following finite-volume analog of Theorem 2.2 is well-known: Theorem 2.6 (ERRW as RWRE on finite boxes, Theorem 3.1 in [Rol03]).Let a > 0 and v 0 ∈ V (N ) .There is a probability measure Q (N ) v 0 ,a on Ω (N ) , such that for all events A ⊆ (V (N ) ) N 0 , one has (2.15) Up to an arbitrary normalization of the random edge weights x ∈ Ω (N ) , the law v 0 ,a of the random environment is unique.
In [Dia88] and [KR00], the distribution Q (N ) v 0 ,a is described explicitly; see also Lemma 4.1, below.The weaker statement that the edge-reinforced random walk on (V (N ) , E (N ) ) is a mixture of Markov chains follows already from [DF80].
The following two theorems are finite-volume analogs of Theorems 2.3 and 2.4.The bounds are uniform in the size of a finite box.
Theorem 2.7 (Decay of the expected log weights in finite volume).There exist functions c 2 , c 3 : (0, ∞) → (0, ∞) with c 2 (a) → ∞ as a ↓ 0, such that the following holds: For all a > 0, all N ∈ N, and all v ∈ V (N ) \ {0}, (2.16) Theorem 2.8 (Decay of the expected weight power in finite volume).For all initial weights a > 0, there are c 1 (a) > 0 and β(a) > 0, such that for all N ∈ N and all v ∈ V (N ) \ {0}, the following holds: (2.17) The analog of Theorem 2.5 holds also in finite volume, uniformly in the size N of the box: Lemma 2.9.For all a > 0 and all α ∈ (0, a/2), one has (2.18) Furthermore, c 4 (a) := sup 3 The big picture -intuitively explained The main work in the paper consists of showing the bounds of Theorems 2.7 and 2.8, which concern finite boxes.Before giving detailed proofs, let us first explain roughly and informally the ideas coming from statistical mechanics.The purpose of this section is only to give an overview and some intuitive picture.Detailed proofs are given in Sections 4-8; they can be checked even when one ignores this intuition completely.
We start Section 4 by reviewing an explicit description of the mixing measure v 0 ,a from equation (2.15), (Lemma 4.1, below).For two different starting points 0 and ℓ of the random walk paths, the mixing measures are mutually absolutely continuous with the Radon-Nikodym derivative We interpolate between Q (N ) 0,a and Q ℓ,a by an exponential family, in other words, by a family of Gibbs distributions with a normalizing constant (partition sum) see Figure 1 and formula (8.10), below.Note that Figure 1: A qualitative picture of log Z η as a function of η.
log Z η η 0 0 1 In the language of statistical physics, the random environment described by P η may be imagined as a rubber membrane, exposed to thermal fluctuations.For a vertex v, 1 2 log x v is interpreted as the height of the membrane at the location v; thus Σ ℓ = 1 2 (log x ℓ − log x 0 ) gets the interpretation of a height difference.
The parameter η may be imagined as an external force: The "membrane" described by P η is exposed to a force η at ℓ and force 1 − η at 0; see formula (4.10), below, and figure 2. We are interested in the elastic properties of the rubber membrane: • On the one hand, this involves how much the expected height difference E Pη [Σ ℓ ] changes as the force η varies from 0 to 1. Indeed, the expected logarithm in (2.16) can be interpreted in terms of this change: • On the other hand, it also involves the physical work (change of the free energy, where the free energy is given by − log Z η ) acting on the membrane as the force η varies from 1/2 to 0. Indeed, the logarithm of the quantity bounded in (2.17) gets the interpretation of this physical work: Both items can be bounded by the same method (Section 6), using a general variational principle for free energies.Let us first explain this variational principle abstractly: Consider two mutually absolutely continuous probability measures, a reference measure σ and a thermal measure µ, where with a Hamiltonian H, a temperature T = β −1 , and the normalizing constant . (3.7) For probability measures ν, define the free energy with the internal energy U (ν) = E ν [H] and the entropy (with the sign convention used in physics) S(ν, σ) = −E ν [log(dν/dσ)], whenever these quantities exist.Then the functional ν → F (ν, σ) is minimized for ν = µ, with the minimum F (µ, σ) = −β −1 log Z. Indeed, We apply this variational principle for 0,a , β = η, H = Σ ℓ , and Z = Z −1 η .In Subsection 5.2, we take some "deformed measure" ν = Π γ,η ((5.13), below) of σ, with a deformation parameter γ.The deformed measure ν is intended to be close to µ.The physical picture of the rubber membrane exposed to external forces motivates us to define the deformed measure ν, using a deformation map Ξ γ (Definition 5.5, below), which changes the heights of the membrane approximately proportional to the Green's function of the Laplace operator in two dimensions (Definitions 5.2 and 5.3, below).With the right choices, the internal energy changes linearly in the deformation parameter, (expectation of (6.5), below), (3.10) while the the entropy fulfills a quadratic bound as it has a minimum at γ = 0.
The proof of this entropy bound with good estimates for the constant const(ℓ) is technically involved (Subsection 5.2).We have much better bounds for the constant const(ℓ) in the case η = 1 than for all other values of η, since P 1 is much easier to control than general P η ; this is the main obstruction against an answer to Diaconis' recurrence question.
By the variational principle, we arrive at Optimizing over γ yields the key estimate (Theorem 6.1 below) In the special cases η = 1 and η = 1/2, this yields the bounds for the expectations in (3.4) and (3.5) as claimed in Theorems 2.7 and 2.8, uniformly in the size of the box (Subsection 8.1).The infinite volume versions, Theorems 2.3 and 2.4, are then derived in Subsection 8.2 from these theorems by taking an infinite volume limit, using tightness arguments from [MR07c].The latter are prepared in Section 7. Finally, in Subsection 8.3, we derive bounds for the hitting probabilities of the edge-reinforced random walk.Entropy bounds for deformed measures Section 5 The key estimate Section 6 Tail estimates for the random environment Section 7 Proofs of the main theorems Section 8 4 The random environment in finite volume Throughout this section, fix a box size N ∈ N and an initial weight a > 0.
First, we state a description of the random environment for G (N ) := (V (N ) , E (N ) ).Let e 0 ∈ E (N ) be any reference edge.Later, from Section 5 on, it will be convenient to choose the edge e 0 adjacent to the origin.So far, the arbitrary normalization of the random edge weights x ∈ Ω (N ) in the representation (2.15) of the edge-reinforced random walk on G (N ) as a random walk in a random environment has not been specified.However, in this section, it is convenient to choose the normalization We introduce a reference measure ρ on Ω (N ) to be the following product measure: Here δ 1 denotes the Dirac measure on (0, ∞) with unit mass at 1. Let T (N ) denote the set of all spanning trees of (V (N ) , E (N ) ), viewed as subsets of the set of edges E (N ) .For a given starting point v 0 ∈ V (N ) of the random walk paths, the distribution v 0 ,a of the random weights can be described as follows: v 0 ,a of the random environment is absolutely continuous with respect to the reference measure ρ with density x a e x e (4.3) with some normalizing constant z (N ) v 0 ,a > 0, not depending on the choice of the reference edge e 0 .
The claim of the lemma is essentially the formula of Coppersmith and Diaconis [CD86] for the distribution of the random environment, transformed such that one has the normalization x e 0 = 1.The transformation to this normalization and thus the proof of Lemma 4.1 is given in the appendix (Section 9.1, below).Now, we consider an interpolation between the random environments Q ℓ,a associated with two different starting points 0 and ℓ in V (N ) .We introduce an "external force" ηΣ ℓ with the switching parameter η ∈ [0, 1].Turning the external force off (η = 0) corresponds to Q (N ) 0,a , while turning the external force completely on (η = 1) corresponds to Q (N ) ℓ,a .More formally, we proceed as follows: Definition 4.2 (Interpolated measures for the environment).For ℓ ∈ V (N ) and 0 ≤ η ≤ 1, we define the following probability measure on Ω (N ) : with some normalizing constant Z (N ) η,0,ℓ,a .This means: η,0,ℓ,a is finite.Note that this definition is independent of the choice of the reference measure ρ, as long as both Q ℓ,a are absolutely continuous with respect to ρ.
We define the random variable Then, the following identity holds: In the formula in (4.8), there appears no normalizing constant, because there is a reflection symmetry of the box V (N ) which interchanges 0 and ℓ.Recall that the box V (N ) has periodic boundary conditions, and that the normalizing constant z (N ) v 0 ,a does not depend on the choice of the reference edge e 0 by Lemma 4.1.Furthermore, using (4.8), note that P (N ) η,0,ℓ,a is absolutely continuous with respect to Q (N ) 0,a with the density Together with (4.3) this implies x a e v∈V (N ) x e .(4.10)

Entropy bounds for deformed measures
In this section, we introduce a deformation map Ξ γ : Ω (N ) → Ω (N ) , which serves to introduce a deformed measure Π γ,η from P η .This deformed measure has the purpose to be a good approximation for Q 0,a in a variational principle, but in such a way that one can estimate its entropy with respect to P η .
From now on, we assume that the reference edge e 0 is adjacent to the origin: e 0 ∋ 0.

The deformation map
We prepare the definition of the map Ξ γ by introducing an approximation D e to the Green's function of the Laplace operator in two dimensions.
Let us first introduce some notation: Recall the definition Ṽ , where we use the convention Ṽ (−2) = ∅.Identifying V (N ) with the subset Ṽ (N ) ⊂ Z 2 , the level of ℓ is also defined for vertices ℓ ∈ V (N ) .Note that the level sets are defined to have width 2 instead of width 1, see Figure 4. Assumption 5.1 (Choice of parameters).In the following, we assume that (iv) ℓ ∈ V (N ) is a vertex on a level l > 3n a .
In the following two definitions, we introduce a modified, truncated version of the Green's function in 2 dimensions.At first, we take the logarithm, appropriately scaled and truncated: Definition 5.2 (Auxiliary function 1: truncated and scaled logarithm).Define a function ϕ = ϕ l,a : for n a ≤ n ≤ l − 1, 1 for n ≥ l.
(5.1) e,0,ℓ,a : Ω (N ) → [0, 1] as follows: Let x ∈ Ω (N ) and l = level(ℓ).Using this l, take ϕ = ϕ l,a from Definition 5.2. (5.2) • Otherwise, we set (5.3) Thus e → D e (x) is an approximation to the Green's function in 2 dimensions, slightly dependent on x for technical reasons.It has the property that for any vertex v not being a corner point, there is at most one edge e ∋ v with D e (x) = ϕ(level(v)).This property is very convenient below, and it is our reason to define the level sets having width 2 instead of width 1.
We write the set E (N ) of edges as a disjoint union  Assume that u and v are on different levels and {u, v} ∩ C (n) = ∅ for all n ∈ N 0 .Then, D e is constant on the sets {x u < x v }, {x v < x u }, and {x u = x v }.Observe that x e ′ . (5.5) Since u and v are on different levels, but not corner points of any box Ṽ (n) , all edges e ′ = e incident to u have endpoints on the same level, namely on level(u), see Figure 6.Similarly, all edges e ′ = e incident to v have endpoints on level(v).In other words, e ′ ≺ e holds for these edges e ′ .Thus all x e ′ appearing in the two sums on the right hand side of (5.5) are e -measurable; recall the definition (5.4) of e .Consequently, because of (5.5), we have With these preparations, we can now introduce the deformation map Ξ γ .Roughly speaking, on a logarithmic scale, the random environment x e is just shifted by γ times the approximate Green's function D e : Definition 5.5 (Deformation).For any γ ∈ R, we define the deformation map Ξ γ = Ξ (N ) γ,0,ℓ,a : Ω (N ) → Ω (N ) by Ξ γ (x) = (exp (γD e (x)) • x e ) e∈E (N ) . (5.6) In the following, we denote the composition of the map Ξ γ : Ω (N ) → Ω (N ) with the map x v : Ω (N ) → (0, ∞), (x e ) e∈E (N ) → e∋v x e for any v ∈ V (N ) by (5.7) The following lemma collects some basic facts about this deformation map Ξ γ .
Lemma 5.6 (Basic facts of the deformation map).With parameters as in Assumption 5.1, for all γ ∈ R, the map Ξ γ is measurable and measurably invertible.Furthermore, it has the following properties: 1.One has (5.8) 2. The reference measure ρ is invariant with respect to Ξ γ .
2. We list the edges in E (N ) in such a way that every e = {u, v} ∈ E (N ) with level(u) = level(v) gets a smaller index than every e ′ = {u ′ , v ′ } ∈ E (N ) with level(u ′ ) = level(v ′ ).Thus, we get a list e 0 , e 1 , . . ., e K with the property that e has a smaller index than e ′ whenever e ≺ e ′ .We rewrite the definitions (5.6) of Ξ γ and (4.2) of the reference measure ρ in logarithmic form: log (x e • Ξ γ ) = log x e + γD e (x), (5.9) (5.10) e -measurable, we see that log x e j is just translated by a value which depends only on the components x e i with i < j.Since 0 ∈ e 0 , we have D e 0 = 0, and the component x e 0 remains unchanged.Such translations leave the reference measure ρ invariant.Here we use that every measurable map f : R d → R d of the form (5.11) leaves the Lebesgue measure invariant.
One verifies that the inverse of Ξ γ is given by (5.12) x ∈ Ω (N ) , e ∈ E (N ) .This is a recursive system for determining the inverse, since D e ([Ξ γ ] −1 (x)) depends only on the components x f with f earlier in the list e 0 , . . ., e K than e.It follows that [Ξ γ ] −1 is measurable.

A quadratic entropy bound
The goal of this subsection is to prove the following lemma: Lemma 5.7 (Quadratic entropy bound).With parameters as in Assumption 5.1, for all γ ∈ R and all η ∈ [0, 1], the image measure γ,0,ℓ,a P (N ) η,0,ℓ,a (5.13) of P η with respect to Ξ γ is absolutely continuous with respect to P η .
Throughout this subsection, we assume that a, n a , N , ℓ, and l are fixed according to Assumption 5.1.
For γ ∈ R and η ∈ [0, 1], we denote by (5.18) the image measure of P η under the inverse of Ξ γ .In the following, we suppress the dependence of D e on the environment x.Thus, we write D e instead of D e (x).We abbreviate for v ∈ V (N ) , γ ∈ R, T ∈ T (N ) , and x ∈ Ω (N ) : x v,γ := In order to estimate the entropy in formula (5.15), we use the following lemma.It provides an explicit formula for the logarithm of the relevant Radon-Nikodym derivative.
Lemma 5.8. 1.For any γ ∈ R and η ∈ [0, 1], we have where "≪" means that the left-hand side is absolutely continuous with respect to the righthand side.
Since D e takes only values in [0, 1], we have e −|γ| x e ≤ e γDe x e ≤ e |γ| x e for all e ∈ E (N ) .Hence, Thus, it follows from (5.21) that x → log(dP η /dΠ − γ,η (x)) is a bounded measurable function on Ω (N ) .Furthermore, using the first equality in (5.21), we see that x → dΠ γ,η /dP η is also a bounded measurable function on Ω (N ) .Consequently, the entropies in (5.22) are both finite.Using Π γ,η = Ξ γ P η , we obtain (5.26) Recall the definitions (5.19).Fix x ∈ Ω (N ) and take v ∈ V (N ) and γ ∈ R. We define a probability measure µ v,γ = µ (5.27) For our fixed x ∈ Ω (N ) , we view , as a random variable on the probability space (E ), again suppressing the dependence on the parameter x in the notation; here P(A) denotes the power set of the set A.
Recall that we want to derive a bound for an entropy proportional to γ 2 .This is done by a second order Taylor expansion with respect to γ.The relevant second derivative is estimated in the following lemma.
The second order Taylor expansion for the entropy is derived in the following lemma.
Proof.Let e = {u, v}.Since x e ≤ x u and x e ≤ x v , we have L e ≤ 1 and the claim follows in the case 0 ≤ η < 1.
Intuitively, equation (5.42) has the following interpretation: We view L 2 e as the probability that the Markovian random walk in the environment x traverses the edge e twice (forward and backward) as soon as the random walker visits one of the two vertices of e.The left hand side of the equation in (5.42) is then the probability of the same event with respect to the mixture of Markov chains.On the other hand, the right hand side in the same equation equals the probability of the same event with respect to the reinforced random walk.
We have now all tools to prove the quadratic entropy bound.
Then, there are two possibilities (see Figure 4): If m is odd, all edges incident to v have both endpoints on level l v .Otherwise, m is even and m = 2l v .In both cases, by Definition 5.3, it follows that D e ′ = ϕ(l v ) for any e ′ incident to v. Hence, we have the estimate: (5.45) Case 2: Assume that v ∈ V (N ) \ {0, ℓ} is a neighbor of a corner point u with level(u) = l v (see Figure 7).
(5.46) Let I denote the set of all vertices v ∈ V (N ) \ {0, ℓ} considered in case 2.Then, (5.47) The factor 8 arises since there are 8 edges connecting corner points at level n to vertices at level n + 1: Each of the 4 relevant corner points at level n is connected to 2 vertices at level n + 1.
Case 3: Assume that v ∈ V (N ) \ {0, ℓ} is not a corner point of any box and v is not a neighbor of a corner point at a different level (see Figure 8).Then, there is precisely one vertex u adjacent to v with l u := level(u) = l v .We set e(v) := {u, v}.
One has D e ′ = ϕ(l v ) for all edges e ′ ∋ v with e ′ = e(v), and thus, it follows: (5.48) Furthermore, since we have excluded v to be as in case 2, the definition (5.3) applies to D e(v) .
In particular, if x v < x u , then D e(v) = ϕ(l v ), and hence (5.49) For e ′ incident to v, we know that the difference D e(v) − D e ′ takes one of the three values 0, ϕ(l u ) − ϕ(l v ), and (ϕ(l u ) − ϕ(l v ))/2.Consequently, (5.50) Assume that x u ≤ x v .Then, combining (5.48) and (5.50) and using (5.51) Because of (5.49), this estimate is also true in the case x v < x u .
A side remark: At this point, it becomes clear why in Definition 5.3, D e was introduced in such a tricky, x-dependent way: If we had used a more naive definition of D e instead, formula (5.51) would have failed to hold.
Integrating both sides of (5.51) with respect to P η and applying Lemma 5.11 gives (5.52) We sum the preceding inequality over the different vertices v: (5.53) The factor 8(4n + 1) arises, since there are not more than 4(4n + 1) edges connecting level n to level n + 1.Each of these edges is counted at most twice, once for each of its two endpoints.

The key estimate
The following theorem is the key to all bounds in the main theorems.One might view it as a bound for the Legendre transform of the logarithm of the partition sum Given the basic facts on the deformation map Ξ γ and the entropy bound from the previous section, its proof boils down to non-negativity of relative entropies.
7 Tail estimates for the random environment The following tail estimates are proved in [MR07c].We need them for tightness arguments when we take the infinite-volume limit in Subsection 8.2, below.Recall that the random environment 0,a for the finite box V (N ) is unique up to a multiplication of the edge weights by a constant.Theorem 7.1 (Theorems 2.3 and 2.4 in [MR07c]).For all a > 0, there are constants c 8 (a) > 0 and c 9 (a) > 0, depending only on a, such that the following estimates hold for all N ∈ N: 1.For all e, f ∈ E (N ) with e ∩ f = ∅, and all M > 0, one has 2. For all v ∈ V (N ) , all e ∈ E (N ) incident to v, and all M > 0, one has Proof of Lemma 2.9.Let a > 0, N ≥ 2, and let e, f ∈ E (N ) satisfy e ∩ f = ∅.Using (7.1), we obtain for any α > 0: The last integral is finite whenever α ∈ (0, a/2).Since the upper bound in (7.3) is uniform in e, f , and N , the claim (2.18) follows.
To prove (2.19), let v, w ∈ V (N ) with |v − w| = 1.Denote by e := {v, w} the edge connecting v and w.Since x e ≤ min{x v , x w }, it follows that log(x v /x e ) ≥ 0 and log(x w /x e ) ≥ 0. Using this and (7.2), we obtain clearly, the upper bound is uniform in v, w, and N .This completes the proof of (2.19).
8 Proofs of the main theorems where the constant c 4 (a) is taken from formula (2.19).Note that c 2 (a) → ∞ as a → 0 because n a → ∞ as a → 0 and lim c 5 (a, 1) < ∞.Now let N ∈ N and v ∈ V (N ) \ {0}, and set l := level(v).We distinguish two cases; first finitely many exceptional cases, and then the general case.
Case 1: N ≤ 6n a or l ≤ 3n a .In this case, there is a path 0 joining the vertices 0 and v of length k ≤ 12n a ; recall that levels have width two.Taking the expectation and using the formula (2.19) from Lemma 2.9 and the fact that |v| ≤ 12n a , we obtain the bound (2.16): Case 2: N > 6n a and l > 3n a .In this case, Theorem 6.1 is applicable with ℓ = v.Using (4.8) and v,a , we rewrite (6.2) for the special value = 1 in the form for all l > 3n a .Recall that the box V (N ) has periodic boundary conditions.Using reflection symmetry, we interchange 0 and v to obtain the claim (2.16):Let N ∈ N and v ∈ V (N ) \{0}, l = level(v).We distinguish the same two cases as in the previous proof: Case 1: N ≤ 6n a or l ≤ 3n a .We observe that E Q (N ) 0,a [(x v /x 0 ) 1/4 ] is bounded: Using (4.8) and z 1/4 ≤ 1 + z 1/2 for all z ≥ 0, we get with the abbreviation (6.1): in the last step, we used ≤ 2 √ 2l ≤ 6 √ 2n a .Case 2: N > 6n a and l > 3n a .This time, we apply Theorem 6.1 with η = 1/2 and ℓ = v.Note that by (4.4), we have dP 1/2 dQ 0 = 1 and thus Using reflection symmetry again, we interchange 0 and v in the following computation: and thus Inserting this in the estimate (6.2) of Theorem 6.1 with η = 1/2, we obtain the claim (2.17) of Theorem 2.8:

Bounds in infinite volume
In this subsection, we deduce the infinite-volume results Theorems 2.3, 2.4, and Theorem 2.5 from their finite volume analogs, namely Theorems 2.7, 2.8, and Lemma 2.9.First, we use a tightness argument given in the appendix to pass to the infinite volume limit.
Proof of Theorems 2.3, 2.4, and Theorem 2.5.Let a > 0, and let v ∈ Z 2 .By Lemma 9.2 in the appendix, there is a measure Q 0,a on Ω = (0, ∞) E and a strictly increasing sequence (n(k)) k∈N of natural numbers with the following properties: a) For any finite subset F ⊂ E, the Q -distribution of (x e ) e∈F converges weakly to the Q 0,a -distribution of (x e ) e∈F .
b) The representation (2.7) of edge-reinforced random walk on Z 2 as a mixture of Markov chains holds with the mixing measure Q 0,a .
Recall from (4.1) that the weights are normalized such that x e 0 = 1 holds Q (n(k)) 0,a -a.s. for a fixed reference edge e 0 ∈ E.
To prove Theorem 2.4, let v ∈ Z 2 \ {0}.Since (x v /x 0 ) 1/4 takes only positive values and is a continuous function of the finitely many weights x e with e ∋ v or e ∋ 0, we conclude we used Theorem 2.8 in the last step.This proves Theorem 2.4.Using (2.18) from Lemma 2.9, the same argument yields Theorem 2.5.
To prove Theorem 2.3, we observe that log(x v /x 0 ) is also a continuous function of the finitely many weights x e with e ∋ v or e ∋ 0. Let 0 = v 0 , v 1 , . .., v L = v be a path from 0 to v, and let Then, for all N large enough that v i ∈ V (N ) for all 1 ≤ i ≤ L and all M > 0, one finds: in the last but one step, we used x e i ≤ min{x v i , x v i−1 }, and in the last step we used (7.2) from Theorem 7.1.Thus, log(x v /x 0 ) has exponential tails, uniformly in the size N of the box.
Consequently, for any fixed vertex v, the law of log(x v /x 0 ) with respect to Q (N ) 0,a is uniformly integrable in N for large N .Abbreviating f M (x) := (x ∧ M ) ∨ (−M ) for the truncation at M and −M , we get, using uniform integrability to interchange limits: The last inequality follows from the bound (2.16).

Hitting probabilities for ERRW
Finally, we apply our bounds for the random environment to deduce estimates for the hitting probabilities for the edge-reinforced random walk.
Proof of Theorem 2.1.1.We claim first that all x ∈ Ω and all v ∈ Z 2 \{0}, the probability Q 0,x [τ v < τ 0 ] for the random walk starting in 0 in the fixed environment x to visit v before returning to 0 and the probability Q v,x [τ 0 < τ v ] for the random walk with exchanged roles of 0 and v in the same environment are connected by the following equation: To prove this claim, take two vertices u = w.Denote by Π u,w the set of all admissible finite paths π = (u = v 0 , v 1 , . . ., v n = w), (8.17) n ∈ N, joining u and w, which visit u or w precisely once, namely at the end points.For any such path π, we introduce the event that π is an initial piece of the random path.Note that the events A π , π ∈ Π u,w , are pairwise disjoint.Furthermore, let denote the reversed path.Note that the reversion defines a bijection • ↔ : Π 0,v → Π v,0 .Moreover, for any path π as in (8.17), Now, we take u = 0 and w = v.Summing (8.20) over all π ∈ Π 0,v , we obtain the claim (8.16) as follows: From this, we conclude Taking the 1/4-th power and expectations yields we used the representation of the edge-reinforced random walk as a random walk in random environment from Theorem 2.2 in the first step and the bound (2.10) from Theorem 2.4 in the last step.This shows part (a) of Theorem 2.1.
2. To prove part (b), let Σ n u,w denote the set of all admissible paths from u to w of length n.Again, reversion yields a bijection between Σ n u,w and Σ n w,u , and the events A π , π ∈ Σ n u,w , are pairwise disjoint.In analogy to (8.21), we obtain Using this, an analogous argument to (8.23) yields the claim (2.2).

Appendix
9.1 Proof of Lemma 4.1 In this appendix, we consider a generalization of Lemma 4.1 to arbitrary finite graphs.It essentially states the formula of Coppersmith and Diaconis [CD86] for the law of the random environment, transformed to a special normalization.
Consider edge-reinforced random walk on any finite graph (V, E) with starting point v 0 ∈ V and initial weights a = (a e ) e∈E ∈ (0, ∞) E .Recall definition (2.8) of x v ; we use the similar notation a v = e∋v a e .For x = (x e ) e∈E ∈ (0, ∞) E , we set Lemma 9.1.The above edge-reinforced random walk on (V, E) has the same distribution as a random walk in a random environment given by random positive weights x = (x e ) e∈E on the edges.Normalizing x such that xe 0 = 1 for a fixed reference edge e 0 , the law of x has the density φ v 0 ,a with respect to the Lebesgue measure δ 1 (dx e 0 ) e∈E\{e 0 } dx e the hyperplane H = {(x e ) e∈E ∈ (0, ∞) E | xe 0 = 1}.
Note that the normalizing constant c 10 (v 0 , a) does not depend on the choice of the reference edge e 0 .
Proof.By Theorem 3.1 of [Rol03], the edge-reinforced random walk on (V, E) has the same distribution as a random walk in a random environment given by random positive weights x = (x e ) e∈E on the edges.The law of the random environment Q ∆ v 0 ,a , normalized such that e∈E x e = 1, has a density with respect to the normalized surface measure on the simplex ∆ = {(x e ) e∈E ∈ (0, ∞) E | e∈E x e = 1}.The density is provided by Theorem 1 in [KR00].Combining this theorem with the matrix-tree-theorem ([Mau76], p. 145, Theorem 3', see also Theorem 3 in [KR00]), it is given by = φ v 0 ,a (x).(9.10) We combine this with (9.3).Using that the projected normalized surface measure πσ has the density (|E| − 1)! with respect to the Lebesgue measure on π[∆], we get that the transformed distribution Q v 0 ,a = F Q ∆ v 0 ,a has the density φ v 0 ,a (x) with respect to the Lebesgue measure δ 1 (dx e 0 ) e =e 0 dx e on the hyperplane H.This proves the claim.

A tightness argument for infinite-volume limits
In this appendix, we prove a variant of Lemma 5.3 of [MR07c] for more general graphs.Consider any infinite, locally finite, undirected, weighted graph G ∞ = (V ∞ , E ∞ , a) with vertex set V ∞ , edge set E ∞ and positive weights a = (a e ) e∈E∞ on the edges.Furthermore, let G N = (V N , E N , a N ), N ∈ N, be a sequence of finite, undirected, weighted graphs with weights a N = (a N,e ) e∈E N ∈ (0, ∞) E N .Finally, let GN = ( ṼN , ẼN , ãN ), N ∈ N, be an increasing sequence of connected weighted subgraphs of G ∞ with the weights ãN induced by G ∞ with the following properties: 1.For any N , GN is a full subgraph of G ∞ , i.e., for every edge e ∈ E ∞ with e ⊆ ṼN , one has e ∈ ẼN .
3. Furthermore, for any N , the graph GN is also a full weighted subgraph of G N .In particular, restricted to ẼN , the weights on G N and on G ∞ coincide.
Take a starting point 0 ∈ Ṽ0 and a reference edge e 0 ∈ Ẽ0 .Consider edge-reinforced random walk on G N and on G ∞ with starting point 0 and initial weights a N and a, respectively.Let Q N denote the distribution of the random environment on (0, ∞) E N in the representation of the edge-reinforced random walk on G N as in Lemma 9.1, normalized such that x e 0 = 1 holds Q N -almost surely.The following lemma is a variant of Lemma 5.3 in [MR07c].In that paper, we consider the special case GN = G N .Boxes with periodic boundary conditions, as needed in our application, are not covered by the cited lemma.

is a decreasing function of a with
still an open problem.

Figure 2 :
Figure 2: An intuitive picture for a rubber membrane described by P η

Figure 3
Figure 3 displays the logical dependence of the different parts of the proofs.

Figure
Figure 3: Overview

Figure 4 :
Figure 4: Vertices at level l are located on the solid lines denote the set of corner points of the box Ṽ (n) .Definition 5.3 (Auxiliary function 2: modified Green's function).For every e = {u, v} ∈ E (N ) , we define a map D e = D (N ) where F denotes the set of all edges e = {u, v} ∈ E(N ) where u and v are on the same level, and F ′ denotes the set of all edges e = {u, v} ∈ E (N ) with u and v on different levels.For e, e ′ ∈ E (N ) , we write e ≺ e ′ if and only if e ∈ F and e ′ ∈ F ′ .LetF (N ) e := σ(x e ′ : e ′ ≺ e)(5.4)denote the σ-field on Ω (N ) generated by the canonical projections on the coordinates e ′ ∈ E (N ) with e ′ ≺ e.Lemma 5.4.For any e ∈ E (N ) , the map D e is F (N ) e -measurable.Proof.Let e = {u, v}.If level(u) = level(v) or {u, v} ∩ C (2n) = ∅ for some n ∈ N 0 , then D e is constant and there is nothing to show.By definition, our levels have thickness 2. Therefore, if {u, v} ∩ C (2n+1) = ∅ for some n ∈ N 0 , then level(u) = level(v) (see Figure5), and this case has already been taken care of.

Figure 5 :
Figure 5: All vertices drawn are on level n.

Figure 6 :
Figure 6: The edge e has endpoints u and v on different levels.The edges e ′ = e with e ′ ∋ u or e ′ ∋ v are drawn with dashed lines.

Figure 7 :
Figure 7: The vertices marked with a square are the neighbors of corner points considered in case 2.

Figure 8 :
Figure8: Corner points are marked with a cross, neighbors of corner points at a different level (as treated in case 2) are marked with a square.The black dots are the vertices at level l covered in case 3.