The boundary of the range of a random walk and the Følner property*

The range process Rn of a random walk is the collection of sites visited by the random walk up to time n. In this paper we deal with the question of whether the range process of a random walk or the range process of a cocycle over an ergodic transformation is almost surely a Følner sequence and show the following results: (a) The size of the inner boundary |∂Rn| of the range of recurrent aperiodic random walks on Z with finite variance and aperiodic random walks in Z in the standard domain of attraction of the Cauchy distribution, divided by n/ log(n), converges to a constant almost surely. (b) We establish a formula for the Følner asymptotic of transient cocycles over an ergodic probability preserving transformation and use it to show that for admissible transient random walks on finitely generated groups, the range is never a Følner sequence unless the walk is a skip-free random walk on Z. (c) For strongly aperiodic random walks in the domain of attraction of symmetric α-stable distributions with 1 < α ≤ 2, we prove a sharp polynomial upper bound for the decay at infinity of |∂Rn|/|Rn|. This last result shows that the range process of these random walks is almost surely a Følner sequence.

is the random subset of G which consists of the sites visited by the random walk up to time n. The case where G = Z d will serve as a motivating and recurring example in this paper; as this group is Abelian we will denote the random walk in this case by The range is a natural object to study and understanding its size and shape is of great interest for a variety of models in probability theory such as random walk in random scenery; see for example [1,16] where the size of the range is shown to determine the leading term of the asymptotic growth rate of the information arising from the scenery. The size of the range and its fluctuations have been extensively treated in the literature starting with the seminal paper [17] where the authors obtained strong laws in the case of the simple random walk on Z 2 , see also [18] and [19]. A central limit theorem for the range was obtained in [24], whereas the case of random walks with stable increments was treated in [28].
More recently, the focus has shifted towards more involved objects, still related to the range. For example [7] studies the entropy of the range, [5,4] its capacity, [37] the largest gap problem in the range [37]; finally [3] and [30] study the boundary of the range, henceforth denoted by ∂R n , that is the sites in the range with at least one neighbour not visited by the random walk. Apart from its intrinsic interest, the motivation for studying the range and its relatives often stems from their relevance in more intricate models; the capacity of the range is of interest in random interlacements (see [35]), whereas the range itself is relevant in the study of random walks in random scenery.
The main focus of this paper is on asymptotic size of the boundary of the range and its relation to the Følner property of the range. We first present our results in the case of recurrent walks.

Recurrent walks in Z and Z 2
The first and last authors first studied the Følner property of the range in [16], where in the case of the simple random walk on Z 2 , it was shown that the range is almost surely a Følner sequence, that is almost surely lim n→∞ |R n (R n + x)| |R n | = 0, for all x ∈ Z d , (1.1) where for two subsets A, B ⊂ Z d , we write A B for the symmetric difference A B := (A ∪ B) \ (A ∩ B).
One can easily see that the Følner property captures the asymptotic relation between the size of the range |R n | and that of its boundary |∂R n | defined by ∂R n := x ∈ R n : ∃y ∈ Z d \ R n , |x − y| = 1 . ∂ v R n := R n \ R n + v .
Using the above notation, the boundary ∂R n can also be written in the form The boundary of the range of a random walk and the Følner property Notice that a similar relation also holds for any countable, finitely generated group G, with S d replaced by the generators of G. In addition, for any a, b ∈ Z d it can be easily shown that |∂ a R n \∂ b R n | ≤ |∂ a−b R n |, |∂ a+b R n | ≤ |∂ b R n | + |∂ a R n |.
In particular (for countable finitely generated groups) the Følner property is equivalent to |∂R n |/|R n | → 0 as n → ∞. The Følner property is important in ergodic theory and information theory as it implies for example that the partial sums of any G-indexed random field along (F n ) satisfies the mean ergodic theorem [13] and the Shannon- McMillan-Breiman Theorem [27,29].
A rate, at least in expectation, is given by [7], who proved for the simple random walk on Z 2 , that E |∂R n | ∼ Cn/ log 2 (n), whereas [30] proved that We strengthen the results mentioned above by obtaining a strong law of large numbers for the boundary |∂R n | of a recurrent random walk in Z 2 , with finite second moments, or of a random walk in Z in the domain of attraction of the symmetric Cauchy law. Before introducing our first main result we give a precise statement of our assumptions.
It is worth mentioning under (A1) the random variables ξ i ∈ Z 2 have zero mean and finite second moments and therefore S n / √ n converges to a Gaussian random vector in R 2 . For example Simple Random Walk in Z 2 satisfies (A1). On the other hand, random walk satisfying assumption (A2) are in the domain of attraction of the symmetric Cauchy distribution, that is if S n := X 1 + · · · + X n , then S n /n → Z, where Z is a real random variable with characteristic function E exp(itZ) = exp(−|t|) We will always assume that the random walk is aperiodic, namely that there is no proper subgroup of Z d containing the support of ξ 1 . By [34,Theorem 7.1] this is equivalent to requiring that φ(t) = 1 for t ∈ [−π, π] d if and only if t = 0. Notice that when ξ i ∈ Z 2 has finite second moments, aperiodicity implies that the covariance matrix Σ in (A2) is non-singular.
Our first main result is the following.
In addition E(|∂ v R n |) ∼ Cvn log 2 (n) and E(|∂R n |) ∼ Cn log 2 (n) as n → ∞. In the cases covered by Theorem 1.1 the random walks are recurrent, since Re [1 − φ(t)] −1 dt = +∞, see [34,II.8.P1]. By [17,24] and [28] we almost surely have |R n | ∼ dn/ log(n) for some d > 0; thus for all nonzero v ∈ Z 2 (or Z in the Cauchy case), there exists a C > 0 such that almost surely for n large enough, providing a sharp quantitative version of the Følner property for these random walks.
We complete our study of recurrent random walks, by considering Z-valued random walks in the domain of attraction of the symmetric, α-stable distribution with 1 < α ≤ 2.
In these cases Le Gall and Rosen showed the range when scaled appropriately converges in distribution to a nontrivial random variable, namely the Lebesgue measure of the , where W α is the symmetric, α-stable Lévy process [28]. Theorem 1.2. Let S n be a recurrent, aperiodic random walk on Z in the domain of attraction of a nondegenerate, symmetric, α-stable distribution with 1 < α ≤ 2. Then for all > 0, almost surely For α = 2 the quantitative upper bound of Theorem 1.2 is optimal in the polynomial term as can be deduced from the case of the simple random walk on Z, see Remark 4.3. Theorem 1.2 shows that for this class of random walks, the range process is almost surely a Følner sequence. This in turn can be used, for example, to greatly simplify the calculation of the upper bound of the relative complexity of the scenery in [1], which is quite technical and is based on dyadic partitions, as Kieffer's Shannon-McMillan-Breiman formula directly applies to the sequence of sets (R n ) n .

Transient walks
Kaimanovich in a private communication asked which groups admit random walks whose range satisfies the almost sure Følner property. Most groups only carry transient random walks. Indeed, the only finitely generated groups on which there are recurrent random walks are the groups which are virtually cyclic or virtually Z 2 , see [38,Theorem 3.24] meaning that there exists H a finite index subgroup of G which is isomorphic to {0}, Z or Z 2 . As for transient random walks the range grows linearly, Kaimanovich's question is naturally linked to the asymptotic size of the boundary of the range of transient random walks; see the discussion in Section 1.1.
Kesten, Spitzer and Whitman's showed that for all random walks, |R n | n → P (S n = 0, for all n ≥ 1) , almost surely, where clearly the limit is positive for transient random walks, see [34, p. 38]. In the case of transient random walks on Z d with d ≥ 1, Okada showed in [30] that if X 1 , X 2 , · · · , is a sequence of i.i.d. random variables in Z d , then almost surely where {S −n ; n ≥ 0} is an independent copy of {−S n ; n ≥ 0}. We take a more general viewpoint and consider random walks on groups. Let G be a discrete group, {S n : n ≥ 0} be a G-valued random walk and R n := {S 1 , . . . , S n }. We first show that lim n→∞ |R n (R n g)| |R n | = c(g), a.s. for all g ∈ G, where c(g) is given explicitly in terms of return probabilities and is thus closely related with the Green function G(g) = ∞ n=0 P (S n = g). In particular, Proposition 5.4 shows that if the Green function of a random walk vanishes at infinity then its range is almost surely not a Følner sequence. Let us say that a random walk is admissible if the walk starting from the origin can reach any point with positive probability. A function f : G → R satisfies lim g→∞ f (g) = 0 if for all > 0, we have |f (g)| < for all but finitely many elements of G. We then establish the following. Theorem 1.3. Let p be an admissible probability measure on a finitely generated group G. Assume that p defines a transient random walk, and that G is not virtually cyclic. Then the Green function G(g) tends to 0 when g tends to infinity.
The condition that G is not virtually cyclic is necessary for the theorem: in Z, the Green function of a non-centered random walk with finite first moment does not tend to 0 at infinity, by the renewal theorem (and this statement can be extended to virtually cyclic groups). It is worth noting that stronger, quantitative results, are available if one restricts the class of groups under consideration, see [38]. For instance, when G is non-amenable the probabilities p n (id G , x) vanish exponentially fast, uniformly in x, from which the result follows readily. On amenable groups, for symmetric walks with finite support or more generally a second moment, one can sometimes use isoperimetric techniques to obtain much stronger results, specifying the rate of decay of p n (id G , x) and of the Green function at infinity. However, in general, there is no hope to get a quantitative version of Theorem 1.3 as we have made no moment assumption; it suffices to consider p such that ξ 1 is at distance 2 2 n of id G with probability 1/n 2 -then the Green function decays at most like 1/ log log d(id G , x). Moreover, surprisingly few tools apply in all classes of groups, regardless of their geometry.
After completing our proof, we learned that part of the statement of Theorem 1.3, namely if G is not virtually Z 2 , can also be deduced from results of Coulhon on isoperimetric inequalities (see [9,Proposition 3.1]). We nevertheless decided to keep our argument for this case also since it is shorter and more elementary.
There is a simple class of transient random walks for which R n is a Følner sequence. Let X 1 , X 2 , . . . be a Z valued i.i.d sequence with finite first moment, positive expectation, and P (X 1 > 1) = 0. Let S n = n k=1 X k . Then for almost all ω ∈ Ω, {R n } ∞ n=1 is an eventually monotone sequence of growing intervals with |R n | → ∞ as n → ∞. Consequently the range process is almost surely a Følner sequence, and |∂R n | remains almost surely bounded. A similar statement holds for walks with S n → −∞ as n → ∞ and P (X 1 < −1) = 0. We say that these walks are the skip-free walks on Z.
With Theorem 1.3 at hand together with a specific study of virtually cyclic groups, we show the following. Theorem 1.4. Let S n be a transient admissible random walk on a finitely generated group G. If it is not a skip-free random walk on Z, then almost surely {R n } ∞ n=1 is not a Følner sequence.
In particular, for all d ≥ 3, the range of all random walks on Z d is almost surely not a Følner sequence. This answers the question of Kaimanovich in full generality for all transient random walks on finitely generated groups. We obtain in particular that for transient random walks, either |∂R n | is eventually bounded, or it grows linearly, with no intermediate behavior. The remainder of the paper is structured as follows. In Section 2 we introduce the necessary notation, collect some preliminary results and set the setting for proving the main results. Also in Section 2 we extend the results of [30] to transient co-cycles in discrete groups. The added generality of considering co-cycles is not strictly necessary, but it comes at no additional technical complexity and we think is of independent interest. The proofs of the main results, along with the relevant definitions and more precise statements, are given in Sections 3, 4 and 5 respectively. The Appendix contains some auxiliary results.

Notation
Given two sequences (a n ) n and (b n ) n of positive numbers we will write a n ∼ b n if lim n→∞ an bn = 1 and a n b n if there exists C > 0 such that for all n ∈ N, a n ≤ Cb n . For x ∈ R, x ∈ Z denotes the lower floor function of x.

Preliminary results
As mentioned in the introduction we will present the results of this Section in the context of group-valued co-cycles. The abstraction to co-cycles is not strictly necessary for the rest of the paper but it does not add to the complexity of the paper and we think is of independent interest.
Let T be an invertible bi-measurable, ergodic, measure preserving transformation of a probability space (Ω, B, P), meaning that P • T −1 = P. In this case we will refer to (Ω, B, P, T ) as a probability preserving transformation. The transformation is ergodic if for all A ∈ B, T −1 A = A implies that P(A) = 0 or P(Ω \ A) = 0. In this section we will make use of the pointwise ergodic theorem: if a measurable f : Ω → R is P integrable and T is ergodic, then for P almost every ω ∈ Ω, Let (G, ×) be a countable discrete group and denote by id G and m G the identity element and the Haar measure of G respectively. A G-valued cocycle is a function F : Z × Ω → G which satisfies the cocycle identity: for all m, n ∈ Z and ω ∈ Ω, F (n + m, ω) = F (m, ω) × F (n, T m ω).
Any measurable function f : Ω → G determines a G-valued cocycle F via In fact the above representation is general; any G-valued cocycle F is of the form F f for f defined by f (ω) = F (1, ω). These cocycles appear in the projection to the second coordinate of the skew product map T f : Ω × G → Ω × G defined by T f (ω, g) = (T ω, gf (ω)).
We would like to point out that there is no general necessary and sufficient criteria for recurrence of cocycles and that it follows from Herman's (unpublished notes) and [20,2] that all amenable groups admit a recurrent cocycle. As the class of amenable groups includes all Abelian groups as well as some groups of exponential growth the latter class is much more diverse than the class of groups of polynomial growth at most EJP 26 (2021), paper 110.

2.
We conclude that there are many groups which admit a recurrent cocycle and all their random walks are transient. The papers [6,31,32,14] provide some sufficient or necessary criteria for recurrence of cocycles. Let R n (ω) := {F (k, ω); 1 ≤ k ≤ n} be the range, or trace, of the co-cycle up to time n. When no confusion is possible we will write R n for R n (ω). For a finite subset A ⊂ G, we write |A| for its cardinality.
The next proposition was proved in Spitzer [34, pp. 38-40] 1 2 for random walks in Z d and the proof of Proposition 2.1 below follows its lines. The case of Random walks on Z d can be recovered by taking Ω = (Z d ) Z , P is a product measure (distribution of an i.i.d. sequence), T is the Bernoulli shift (ergodic) and f (ω) = ω 0 , where ω = (· · · , ω −1 , ω 0 , ω 1 , · · · ). Proposition 2.1 (pp. 38-40 [34]). Let (Ω, B, P, T ) be an ergodic, probability-preserving transformation, G a countable group and F : Z × Ω → G a cocycle. Then for P-almost every ω, Counting each z ∈ R n according to the last time it has been visited in [1, n] ∩ N, it follows that and for all n ∈ N and N < n n k=1 By the pointwise ergodic theorem, dividing all sides of the inequality by n, one has that for all N ∈ N, for almost every ω ∈ Ω, Noting that A N ↓ A as N → ∞ and thus P (A N ) − −−− → N →∞ P(A) the conclusion follows.
The following corollary is almost immediate. 1 Although it is stated for Z d , it holds for general countable groups as pointed out by Flatto [19]. 2 Spitzer attributes the proof to an unpublished manuscript with Kesten and Whitman.
We need to show that P(l = 0) > 0. Letting B N = {ω ∈ Ω; l = N }, the transience of F implies that there exists N ∈ N ∪ {0} such that P (B N ) > 0. For all ω ∈ B N and j ∈ N, The latter implies that T N B N ⊂ {ω ∈ Ω : l = 0} and since T preserves P we have P(l = 0) > 0.

2.1
The asymptotic size of the boundary of transient cocycles [7] and [30] studied the asymptotic size of the boundary of a random walk in Z d . In the case of transient random walks on Z d , Okada obtained a strong law of the boundary of the range [30]. The following is a similar result for transient cocycles on arbitrary countable groups. The proof follows closely that of [30,Theorem 2.1]. Note that the proof [30, Theorem 2.1] contains a small issue since the shift operator as defined in the beginning of [30, Section 3.1] is measure-preserving with respect to an infinite measure. This however can be easily fixed by using instead the setup mentioned just before Proposition 2.1.

The Følner property
A sequence {K n } n∈N of subsets of G is a right Følner sequence in G if for all n ∈ N, K n is a nonempty finite set and for all g ∈ G, The existence of Følner sequences is equivalent to amenability of the group (existence of a right invariant mean on G). In [16], it was shown that the range of the symmetric random walk in Z 2 is almost surely a Følner sequence. Kaimanovich, in a private communication, asked which random walks on general countable groups have almost surely Følner ranges and Dolgopyat suggested to generalise this question to which cocycles have almost surely Følner ranges. The following is a partial advance on Dolgopyat's question for transient co-cycles as it establishes a relatively simple criterion for checking the Følner property. This criterion will be used afterwards to give a complete answer to Kaimanovich's question.
Proposition 2.5. Let (Ω, B, P, T ) be an ergodic probability preserving transformation, G a countable group and F : Proof. It is easy to see, by multiplying by g −1 , that for all n ∈ N, Since (R n (R n g)) = (R n \ (R n g)) ((R n g) \ R n ), the result follows from Corollary 2.4.

Proof of Theorem 1.1
In [7] it was shown that in the case of the symmetric random walk on Z 2 , E |∂R n | is proportional to the entropy of the range (at time n) and it is of order constant times n/ log 2 (n). [30] has shown that 2 −1 π 2 ≤ C ≤ 2π 2 . The previous Theorem is a law of large numbers type result for a more general class of random walks which includes random walks in the normal domain of attraction of a symmetric Cauchy distribution. It also gives a more precise estimate on the almost sure Følner property of the range of such random walks; see [16] for some consequences of the Følner property of the range in the model of random walks in random sceneries. The proof goes by first establishing an upper bound for the variance of |∂R n | which gives convergence in probability. After that we use a method from [19] which improves the asymptotics of the rate of convergence in probability, thus enabling us to use the Borel-Cantelli lemma for showing convergence almost surely.
To prove the statement in Theorem 1.1 for ∂R n , we need first the statement for ∂ v R n and in fact a more general statement for ∂ V R n , where V ⊂ Z d . The proof of the statement for ∂R n will then follow from the inclusion-exclusion principle. The boundary of the range of a random walk and the Følner property

Auxiliary results
We now state and prove a number of auxiliary results that will be used in the proof of Theorem 1.1, that we could not find in the literature.
Remark 3.2. Proposition 3.1 in the case of symmetric random walks on Z 2 with finite variance was (essentially) treated in [30].
Proof. The proof is a simple application of [26,Theorem 4a]. This theorem states that for any aperiodic random walk in Z d , W ⊂ Z d a finite subset, and any x ∈ Z d we have where in the cases of interest to us, by [ Under assumptions (A1) or (A2), it follows from the local limit theorem that n j=0 P(S j = 0) ∼ c log n. (3.4) Using this asymptotics, a standard argument, see for example [17] and [23, shows that there exists a constant γ d > 0, depending on the random walk, such that Under our assumptions, we have a(x) > 0 for all x = 0, by [ .
Assume first that (A1) holds, i.e., d = 1, and that the walk is strongly aperiodic, which is equivalent to |φ(t)| < 1 for all t ∈ (−π, π) d \ {0}. For aperiodic random walks, this amounts to the condition that the greatest common divisor of return times to the origin is 1. For instance, the simple random walk is aperiodic, but not strongly aperiodic.
By strong aperiodicity of the walk condition (6.6) in [25] is satisfied. By [25, Theorems 8 and 9] we have which implies the result. Under (A2), still with strong aperiodicity, the estimate random variables, independent of the random walk, with P{B j = 1} = ρ ∈ (0, 1) and P{B j = 0} = 1 − ρ. It can be easily checked that {S j } j≥0 is then strongly aperiodic and therefore that there exists C > 0 such that for all n ∈ N, In addition let EJP 26 (2021), paper 110. As Z d is an Abelian group, in this case we consider instead a bi-infinite i.i.d. sequence {X i } i∈Z and write for n ∈ N, In what follows we will make use of the fact that for k < n and v ∈ Z d , for all ω ∈ Ω, , as n → ∞.
In addition, there exists M > 0 such that Using the inclusion-exclusion principle and enumerating the elements of S d as v 1 , . . . , v 2d we have whence we can write as in the proof of Proposition 2.3 To see why notice that Let us first consider the case V = {v}. Then one has . By the Markov property for the random walk, for all 1 ≤ k ≤ n, A k (k) and B k (n − k) are independent, thus 1. An easy calculation, using the fact that log is slowly varying, shows that where we write x for the integer part of x.
Assume now |V | > 1. For any finite set W , the proof of Proposition 3.1 and in particular (3.2) imply that, as n → ∞ (3.5) and (3.3) respectively. Therefore similar arguments show that for any V ⊂ S d we have that and wheng V (0)g V ∪{0} (0) = 0, Going back to ∂R n , we thus have that where the last two inequalities follow from (3.9) and Proposition 3.1. This proves (3.7). Note that this equation gives a semi-explicit formula for the constant in the asymptotics of E |∂R n | in (3.7). We do not know any simpler formula for this constant ( [7] only provides upper and lower bounds, but no exact formula).
Variance. For the second part, from (3.10) we have that Similarly to the proof of (3.7), we first work out in detail the case V = {v}, for v ∈ S d arbitrary. First notice that The boundary of the range of a random walk and the Følner property The first term is equal to E (|∂ v R n |). For the second term, notice that for 1 ≤ k < m ≤ n, since for any k, m, B k (·), A m (·) are decreasing sequences of sets. To keep notation concise, for any integers l, k we will denote P(A k (l)) by ψ(l) and P(B k (l)) by θ(l), where we can drop the dependence on k since P • T −1 = P. Since for 1 ≤ k ≤ m ≤ n the events are independent, we have This shows that for k < m, it follows that for all (k, m) ∈ D(n), and .
EJP 26 (2021), paper 110. By Lemma 3.3, the term in the sum is bounded by C/(j log 2 (j)). Thus, since (m − k)/2 → ∞ when n → ∞, and similarly whence |∂R n |/ E |∂R n | → 1 in probability. In particular, from (3.7), there exists C > 0 such that log 2 (n)|E(|∂R n |)/n → C, and thus by elementary arguments In order to prove Theorem 1.1 we would like to improve the convergence in probability result to almost sure convergence. The gap of order log log(n)/ log(n) in the decay of these probabilities is enough to guarantee almost sure convergence of |∂R N k | for N k = exp (n a ) with a > 1. By a different method looking at with a > 1/2. This is since for a > 1/2, Unfortunately as |∂R n | is not necessarily monotone, this subsequence is too thin in order to interpolate the almost sure convergence from the subsequence to almost sure convergence along the whole sequence. To that end we use a method from [19].
(ii) The sequence of random variables {X n ; n ≥ 0} satisfies property D(δ) if for all > 0, (3.23) The proof of Theorems A and B is similar to the proof of [19,Thm. 4.2], hence it is postponed to the appendix. We note that the proof of Theorem B is easier for each individual term |∂ V R n | separately then for |∂R n | which is a weighted sum of Proof of Thm. 1.1. Notice that if we manage prove that log 2 (n)|∂R n |/n → C almost surely, then the fact that C > 0 follows from Corollary 3.5. In addition, similarly to the proof of Corollary 3.5 it suffices to prove that for any V ⊂ S d we have By (3.13) it further suffices to show that Let > 0 and write N k := exp 4 √ k . By Corollary 3.8 there exists C > 0 such that for all n ≥ 2, It follows from the Borel-Cantelli lemma that for almost every w ∈ Ω, lim sup k→∞ Z N k ≤ .
As Z n ≥ 0 and is arbitrary it follows that for almost every w ∈ Ω, lim n→∞ Z N k = 0.
Now for a general n ∈ N there exists a unique m = m(n) ∈ N such that N m ≤ n < N m+1 . Since e x − 1 ≤ 2x for all 0 ≤ x ≤ 1 and for all m large 4 √ m + 1 − 4 √ m ≤ 1/3m −3/4 , it follows that for all large m, (3.24) for all m large enough and N m ≤ n ≤ N m+1 , we have using (3.24) Since |∂ V R Nm |/E(|∂ V R Nm |) tends almost surely to 1, we obtain that |∂ V R n |/E(|∂ V R Nm |) also tends to 1. Together with (3.25), this gives |∂ V R n |/E(|∂ V R n |) → 1 almost surely as required. It then follows from (3.13) that for all V ⊂ S d , there exists C V ≥ 0 such that almost surely, and consequently, almost surely as required.

Proof of Theorem 1.2
Let {X n } ∞ n=−∞ be as sequence of i.i.d., centered, Z-valued random variables, and define the two-sided random walk {S n } n∈Z as follows, S 0 = 0, and for n ≥ 1 let S n = X 1 + · · · + X n and S (−) n = −X −1 − · · · − X −n . We assume that {S n } n is aperiodic in the sense of Section 3 and that the random variables {X i } i belong to the domain of attraction of a non-degenerate, symmetric, α-stable distribution with 1 < α ≤ 2. This implies that the X i are centered, and there exists a positive, slowly varying (at ∞) function L : R + → R + such that, where Z α is a real random variable with characteristic function E(e itZα ) = e −|t| α

. By
Lévy's continuity theorem, writing φ(t) := E(e itX1 ), we see that for all t > 0,  If A ⊂ Z is a finite subset we define and write r n := r n (0, {0}).
We will need the following result.  2). Assume in addition that the corresponding random walk is aperiodic. Then r n is regularly varying of index 1/α − 1 as n → ∞; that is there exists another positive slowly varying (at ∞) function M such that, for any x ∈ Z and any finite nonempty A ⊆ Z,  Proof. When lim s→∞ L(s), where L is the first slowly varying function in the statement, exists this result is [25,Theorem 8]. So we will consider the case where L(·) does not have a limit at infinity. Similarly to the proof of [25,Theorem 8] we define for λ ∈ [0, 1) First we will study the asymptotic behaviour of U (λ) as λ → 1.
We claim that the functionŨ is regularly varying of index 1/α − 1 when δ → 0+ and that its asymptotic behaviour at 0+ does not depend on . Let us prove this claim.
Having proved the claim, let us now prove the main result. We write We claim that the term J 1 ( , λ) on the right dominates the other ones when λ tends to 1. First, it dominates the last one asŨ (1 − λ) tends to infinity while the last term remains bounded. The other two terms are similar, so let us handle the first one. We write the difference as dt.
In particular for 0 < t < we can write for some positive function η(·) such that η( ) → 0 as → 0+. Therefore We can similarly bound the other term and since we have shown above that lim δ→0Ũ (δ)/Ũ (δ) = 1 we also have that lim sup Since > 0 is arbitrary and η( ) → 0 with → 0 this proves that λ)), and since r n is monotone it follows that r n ∼ cn 1/α−1 M (n) for some constants c, C > 0 and a positive, regularly varying function M , by [8,Corollary 1.7.3].

Remark 4.2.
The following alternative approach using a local limit theorem is possible. Firstly, by [21], where m ∈ Z and a n is a 1/α regularly varying sequence. Then a similar argument as in [17] shows that r n (m) := P (S k = m for all 1 ≤ k < n) ∼ r n (0) as n → ∞. We claim that this is the optimal rate in the polynomial exponent. Indeed, for the simple random walk on Z, R n is an interval and thus |∂R n | = 2. By Theorem 3 in [11], Before proceeding with the proof of Theorem 1.2, we need a number of auxiliary results. Proposition 4.4. Let S n be an aperiodic random walk on Z in the domain of attraction of a nondegenerate, symmetric, α-stable distribution with 1 < α ≤ 2, then for any > 0, k ∈ N and x ∈ Z \ {0} there exists M > 0 such that Proof. Let > 0 and j ∈ Z \ {0}. In the course of the proof C will denote a global positive constant whose value can change (increase) from line to line. We have the bound where for y ∈ Z and n ≥ 0 we define We will now show that for all k ∈ N there exists C > 0 such that, The latter inequality proves the proposition by a simple induction argument. Writing For every (m 1 , ..., m k ) ∈ Λ k (n) there exists a minimal 1 ≤ l < k such that there exists a minimal l > l with |m l − m l | ≤ 2. Since for all 1 ≤ l < k It remains to bound the other term. For (m 1 , ..., m k ) ∈ ∆ k (n), letting q 1 = m 1 , q k+1 = n − m k and for 2 ≤ j ≤ k, q j = (m j − m j−1 )/2 , where x denotes the integer part of x, we deduce the inequality by replacing the restriction that S j + x = S m l for j ∈ [1, n] by the weaker one that S j + x = S m l for j ∈ [m l−1 + q l , m l + q l+1 ]; see Figure 1 where the random walk started  The bound above is essentially a product of events that depend on non-overlapping sequences of the random variables {X j } j∈Z and therefore by independence we have that where we write C(x) for a generic positive constant depending on x. In particular since L ± (·) are slowly varying, for any > 0 we can find a positive constant C > 0 such that L ± (n) ≤ Cn and thus The sum is restricted to the values of q i that can be produced by the above process. They satisfy q 1 + 2q 2 + · · · + 2q k + q k+1 ≤ n and q 1 + 2q 2 + · · · + 2q k + q k+1 ≥ n − k, because of the integer parts in the definition of q j . We have that  The result follows with = 2 .
Lemma 4.5. Let k ∈ Z, k > 1, and suppose that α 1 , . . . , α k > −1. Then Proof of Lemma 4.5. We will proceed by induction on k. Let k = 2 and notice that q1+q2=n q α1 where β (α 1 , α 2 ) is the value of the β-function at (α 1 , α 2 ). Suppose now that the result holds for all integers ≤ k and notice that where the last two inequalities follow from the inductive hypothesis for j = k and j = 2 respectively. Proposition 4.6. Let S n be an aperiodic random walk on Z in the domain of attraction of a nondegenerate, symmetric, α-stable distribution with 1 < α ≤ 2. Then for all > 0, almost surely lim n→∞ |R n | n 1 α − = ∞. Proof. Let > 0. Le Gall and Rosen have shown in [28] that there exists a 1/α-regularly varying sequence a n such that |R n | a n dist. (4.5) where W α [0, 1] is the range of the symmetric α-stable Lévy motion up to time 1. It is well known, see for example [10], that the occupation measure of a one-dimensional α-stable  Since a n is 1/α-regularly varying, setting t n := n κ , where κ > 0, we have that a tn is κ/α-regularly varying. Let κ = 1 − α /2, so that κ/α > 1/α − . Then for n large enough we have a tn > n 1/α− . Decompose the interval [0, n] ∩ Z into sub-intervals [jt n , (j + 1)t n ], j = 0, 1, .., n/t n of length t n , plus perhaps a remainder interval which will be ignored. Writing R(n, m) = {S n+1 , .., S m }, for δ > 0 and all n large enough we have that |R n | ≥ max j∈{1,.., n tn } |R (jt n , (j + 1)t n )| , and therefore P |R n | < δn By (4.6) we can choose δ > 0 small enough so that P (Leb R (W α [0, 1]) < δ) < 1 and by (4.5) we can choose N 0 large enough so that for all n ≥ N 0 we have P (|R tn | < δa tn ) ≤ ρ < 1.
Therefore for all n > N 0 , from the definition of t n and the above it follows that Since n/t n ∼ n κ where κ = α/2, we have that n≥1 P |R n | < δn  First note that Let i ∈ {1, 2}. By Proposition 4.4 for any > 0 and k ∈ N there exists M > 0 such that for all n ∈ N and i ∈ {1, 2}, Therefore, choosing < δ, there exist k ∈ N and M > 0 such that for all n ∈ N and i ∈ {1, 2},

Følner property and transient random walks
We now turn to an application of Proposition 2.5 in the context of random walks in groups.
Let G be a countable group. Given p a probability measure on G and ξ 1 , ξ 2 , ... an i.i.d. sequence with marginals distributed as p, let S n = ξ 1 · ξ 2 · . . . · ξ n be the corresponding random walk and R n := {S 1 , ..., S n } be its range process. Let us say that a random walk driven by a measure p on the group G is admissible (or irreducible) if the walk starting from the identity can reach any point in the group, i.e. the semigroup generated by the support of p is the whole group. This is a natural irreducibility assumption.
We The next Lemma is well known and we only include its proof for completeness.
Lemma 5.1. If S n is a G-valued transient random walk then for all g ∈ G, Proof. Let g ∈ G. By the Markov property for the random walk, P(L(g) = 0) = q(g) and thus The ergodic theoretic model of the random walk is the following skew product transformation. Let Ω := G Z , P = p ⊗G the product measure on Ω with marginals distributed as p and T : Ω → Ω the full shift defined by (T ω) n = ω n+1 . The dynamical system, G Z , B, p ⊗G , T is a stationary Bernoulli shift, hence ergodic, see for example EJP 26 (2021), paper 110. [15, p. 180]. Writing m G for the Haar measure of G, and f : Ω → G, f (ω) := ω(0), the skew product transformation T f : here π G (ω, h) = h is the projection to the G coordinate. The advantage of working with the skew product is that the cocycle identity indicates what is the relevant random walk in inverse time. In this case, write for n ∈ Z, Corollary 2.4 gives the following extension of Okada's result.
Corollary 5.2. Let G be a discrete countable group and p a probability measure on G and ξ 1 , ξ 2 , ... an i.i.d. sequence with marginal p. Then for any g = id G , almost everywhere lim n→∞ |R n (R n · g)| |R n | = P ∀n ∈ N, S (−) n = g P ( ∀n ∈ N, S n = g| ∀n ∈ N, S n = id G ) Proof. This is a direct consequence of Corollary 2. = g n P ( ∀k ∈ N, S k = g n | ∀k ∈ N, S k = id G ) = 1, and an application of Corollary 5.2 shows that for all large n ∈ N, almost surely We conclude that almost surely the range is not a Følner sequence.
In order to show the second part let g be of infinite order. Consequently, g n − −−− → n→∞ ∞ and there exists n ∈ N such that almost everywhere, By the triangle inequality for cardinality of symmetric differences of sets, for all k ∈ N, since for all h ∈ G and A ⊂ G, |Ah| = |A|. We conclude that for all k ∈ N, Taking limits as k → ∞ we see that almost everywhere, where the last inequality is (5.3). This concludes the proof of the second part.
Corollary 5.5. Let G be a torsion free countable group. Let S n be a G valued transient random walk for which the Green function tends to 0 at infinity. For all g ∈ G \ {id}, there exists c(g) > 0 such that a.e.

Transient random walks on virtually cyclic groups
Let us show that the skip-free random walks (defined just before Theorem 1. is almost surely a Følner sequence then S n is a skip-free random walk. Proof. If E (|X 1 |) = ∞ then by [34, P8, page 287] the Green function of S n and S (−) n decays at infinity. By Corollary 5.5, lim n→∞ (|R n (R n + 1)| / |R n |) > 0. It remains to treat the case when E (|X 1 |) < ∞. If E(X 1 ) = 0, it follows from the Chung Fuchs Theorem [12] that the walk is recurrent, which we exclude. Therefore, by the strong law of large numbers, either lim n→∞ S n = ∞ almost surely or lim n→∞ S n = −∞ almost surely according to whether E(X 1 ) > 0 or E (X 1 ) < 0. It remains to show that if S n → ∞ almost surely and P (X 1 > 1) > 0 then the range process is almost surely not a Følner sequence, the opposite case being similar.
Now if the random walk is transient and and S n → ∞ almost surely then 4 P (∀n ∈ N, S n > 0) > 0.
As the distributions of S Let us show that, on groups which are virtually Z but not Z, there is no transient walk for which the range is Følner. Proposition 5.7. Let S n be a transient random walk on a group which is virtually Z, but not isomorphic to Z. Then the range {R n } ∞ n=1 is almost surely not a Følner sequence.
Proof. By [33,Theorem 5.12], there is a surjective morphism π from G to either Z or the infinite dihedral group Z Z/2, with finite kernel F . Since the kernel is finite, the image under π of the transient random walk on G is still a transient random walk in the image group. We will treat separately the two cases.
Assume first that π(G) = Z. Let p denote the measure on G driving the random walk S n , and p π its image in Z driving the image random walk S π n . Write G, resp. G π for the Green function of S n , resp. S π n . If X π 1 has no first moment, then the Green function of S π n decays at infinity, by [34, P8, page 287]. Since G(g) ≤ G π (π(g)), it follows that the Green function of S n also decays at infinity, since π has finite kernel. Then Proposition 5.4 shows that R n is not a Følner sequence. If X π 1 has a first moment, then E(X π 1 ) has to be nonzero by [12] since the walk S π n is transient. In this case, by the strong law of large numbers, the probability that S π n = 0 for all n ∈ Z \ {0} is positive, as in the proof of Proposition 5.6. Since we are assuming G = Z, the kernel F is nontrivial. Therefore, we may choose g ∈ F \ {id G }. With positive probability, we have S n = g for all n ∈ Z \ {0}, as this follows from the fact that S π n = 0 = π(g) for the projected random walk. Then Corollary 5.2 yields that |R n (R n · g)| /|R n | converges to a positive limit, and therefore R n is not Følner.
Consider now the case where π(G) = Z Z/2. The dihedral group Z Z/2 is made of two copies of Z that we denote by Z and Z × {ρ} where ρ is the nontrivial element of Z/2, acting on Z by sign reversal. Decomposing p π as a sum of two measures on these two copies of Z, we write p π = αp 1 + (1 − α)ρ * p 2 for some probability measures p 1 , p 2 on Z and some α ∈ [0, 1) (the value α = 1 being excluded as the walk is admissible, so it can not remain stuck in Z). Letp i be the reversed measure, given byp i (n) = p i (−n). The first return of the random walk S π n to Z is distributed according to the measure where the k-th term corresponds to trajectories of the random walk jumping to the second copy of Z, making k steps there, and then coming back. Here, * denotes the convolution of measures, i.e., µ 1 * µ 2 is the distribution of the sum of two independent random variables distributed respectively as µ 1 and µ 2 . The presence of the reversed measures is coming from the action of ρ that reverses signs. If p 1 or p 2 has an infinite moment, then so does the measure p . Therefore, the Green function associated to p tends to zero at infinity on Z, by [34, P8, page 287]. Since, for points in Z, it coincides with the Green function of the random walk S π n , it follows that G π (g) also tends to zero at infinity along Z. As G(g) ≤ G π (π(g)), we get a sequence tending to infinity in G along which G(g n ) and G (−) (g n ) tend to 0. By Proposition 5.4, R n is almost surely not a Følner sequence.
Assume now that both p 1 and p 2 have a finite moment. Then so does p . Moreover, the expectation of a variable distributed according to p is given by α k (E(p 2 ) + kE(p 1 ) + E(p 2 )).
As E(p i ) = −E(p i ), this reduces to Therefore, by [12], the random walk associated to p is recurrent. Then so is S π n . This is a contradiction, concluding the proof.

Green functions vanish at infinity
The proof of Theorem 1.4 will essentially follow from Proposition 5.4 and the results of Subsection 5.2, once we establish the decay of the Green function of any admissible transient random walks on non-virtually cyclic groups. This is the main goal of this section.
As in Section 5.1 we consider a probability measure p on a group G, and let S n = ξ 1 · · · ξ n be the corresponding random walk, where ξ i are i.i.d. random variables distributed as p. When this random walk is transient, the Green function G(g) of the walk is defined as in (5.1), by G(g) = ∞ n=0 p n (id G , g), where p n (id G , g) = P (S n = g). More generally, let G(g, h) = G(g −1 h). This is the average time that the walk starting from g spends at h.
Conditioning on the position of the walk at time 1, one gets p 2n+1 (id G , g) ≤ p 2n (id G , id G ). Therefore, for any g and any N , The right hand side is the tail of the converging series n p n (id G , id G ) = G(id G ). If N is large enough, it is bounded by an arbitrarily small constant . For each n < 2N , the measure p n (id G , ·) is a probability measure on G. Hence, there are only finitely many points g for which p n (id G , g) > /(2N ). Summing over n ≤ 2N , it follows that for all but finitely many points one has n<2N p n (id G , g) ≤ , and therefore G(g) = n p n (id G , g) ≤ 2 for all but finitely many points.
The next lemma is the main step of the proof of Theorem 1.3. Lemma 5.10. Consider a finitely generated group G, with a finite symmetric generating set S. Assume that the simple random walk on G (driven by the uniform measure on S) is transient. Let c > 0. Consider a probability measure p on G with p(s) ≥ c for all s ∈ S. Then its Green function tends to 0 at infinity.
When the simple random walk is transient, all the admissible random walks on G are transient, by [38,Theorem 3.24]. Hence, under the assumptions of the lemma, the Green function associating to p is finite, and it makes sense to ask if it tends to zero at infinity.
Proof. The proof will rely on a classical comparison lemma, making it possible to relate general random walks to symmetric ones. Denote by G the Green function associated to p, and by G S the Green function associated to the simple random walk. Under the assumptions of the lemma, [36, Proposition, Page 251] ensures that for any nonnegative square-integrable function f on G, Let F (g, h) be the probability to reach h starting from g. By Lemma 5.1, we have F (g, h) = qG(g, h) for a fixed q > 0, independent of g and h.
Let Ω = G N be the space of all possible trajectories, endowed with the probability measure P corresponding to the distribution of the random walk given by p starting from id G . A cylinder set is a set of the form [g 0 , . . . , g n ] = {ω ∈ Ω | ω 0 = g 0 , · · · , ω n = g n } ⊆ Ω.
Assume by contradiction that G(g) does not tend to 0 at infinity. Then one can find > 0 and an infinite set I ⊆ G on which F (id G , g) = qG(g) > . Let M > 0 be large enough (how large will be specified at the end of the argument). We define a sequence h n of elements of I as follows.
• First, take h 0 = id G . Let also T 0 = [id G ] ⊆ Ω be the set of all trajectories starting from id G , and R 0 = {id G }.
• Then, take an h 1 ∈ I at distance at least M of h 0 . As the probability to reach h 1 starting from id G is > by definition of I, one can find (by throwing away very long trajectories or very unlikely trajectories in finite time) a finite number of cylinder sets starting at id G and ending at h 1 with total probability > . Denote this set of trajectories by T 1 , with P(T 1 ) > . Let R 1 be the set of points that trajectories in T 1 reach before h 1 . As T 1 is a finite union of cylinders, R 1 is finite.
• Then, take an h 2 ∈ I at distance at least M of h 0 and h 1 . We can also require that it does not belong to R 0 ∪ R 1 , as this set is finite while I is infinite. As above, we can then define a set T 2 which is a finite union of cylinders ending at h 2 with P(T 2 ) > , and R 2 the finite set of points reached by these trajectories before h 2 .
• The construction goes on inductively to define h n .
The point of the previous construction is that for i ≤ j we have the inequality Indeed, this is clear for i = j. For i < j, note that trajectories in T i ∩ T j reach h i and then h j , in this order as h j / ∈ R i . Therefore, where the central equality follows from the Markov property. This proves (5.5).
Let us now take N large, and apply the inequality (5.4) to the characteristic function of {h 0 , · · · , h N −1 }. We obtain We will bound the left hand side from below and the right hand side from above to get a contradiction. Thanks to (5.5), we have The Green function G S tends to 0 at infinity, by Lemma 5.9. As the distance between h i and h j is at least M for i = j by construction, it follows that G S (h i , h j ) ≤ η(M ) where η tends to 0 with M . We obtain Combining these two estimates with (5.6) yields We obtain a contradiction by taking M large enough so that c −1 η(M ) < q −1 2 /2, and then letting N tend to infinity. Proof of Theorem 1.3. The result follows from Lemma 5.8 if G is virtually Z 2 . Hence, we can assume that this is not the case, and therefore that the simple random walk on G is transient by [38,Theorem 3.24].
The Green functions for the probability measures p and (p + δ id G )/2 are related by the identity G (p+δ id G )/2 (g) = 2G p (g), by [38,Lemma 9.2]. Without loss of generality, we can therefore replace p with (p + δ id G )/2 and assume p(id G ) > 0. As p is admissible, it follows that there exists N with p N (id G , s) > 0 for all s in the generating set S. By Lemma 5.10, the Green function associated to p N , denoted by G N , tends to 0 at infinity.
To compute G(g), split the arrival times to g according to their values modulo N . For times of the form i + kN , such arrivals can be realized by following p for i steps, and then p N for k steps. It follows that When g tends to infinity, the first term tends to 0 as this is a finite sum and G N tends to 0 at infinity. For large enough g, we get G(g) ≤ 2 G N L ∞ .
We are now ready to complete the proof of Theorem 1.4.
Proof of Theorem 1.4. For groups which are not virtually Z, the result follows directly from Proposition 5.4 and Theorem 1.3 showing that the Green functions G and G (−) tend to 0 at infinity. Groups which are virtually Z but not Z are handled in Proposition 5.7. Finally, the case of Z is done in Proposition 5.6.

As before
∂ v X n,i := X n,i \ {X n,i + v} , |∂X n,i |.
As 0 is arbitrary this concludes the proof of Theorem A.
Proof of Theorem B. The proof essentially follows Sections 5,6 in [19], which is quite lengthy and technical. Rather than reproducing the full argument we opt instead to point out the necessary modifications for the statements of the relevant results in [19] to apply to the objects of interest in our case. Divide the random walk path S = {S j ; 1 ≤ j ≤ n} into N = [n/L] + 1 blocks of size L = [n/ log log n], where [x] denotes the integer part of x. We will write S (i) for the i-th block and R Finally the object T ij must be replaced by T i,j := x ∈ ∂ V R (i) n : R (j) n ∩ (x + (V ∪ {0})) = ∅ .
One can similarly modify the proof of [19,Theorem 5.2].