Recurrence and transience for long-range reversible random walks on a random point process

We consider reversible random walks in random environment obtained from symmetric long--range jump rates on a random point process. We prove almost sure transience and recurrence results under suitable assumptions on the point process and the jump rate function. For recurrent models we obtain almost sure estimates on effective resistances in finite boxes. For transient models we construct explicit fluxes with finite energy on the associated electrical network.


Introduction and results
We consider random walks in random environment obtained as random perturbations of long-range random walks in deterministic environment.Namely, let S be a locally finite subset of R d , d 1 and call X n the discrete time Markov chain with state space S which jumps from a site x to another site y with probability p(x, y) proportional to ϕ(|x − y|), where ϕ : (0, ∞) → (0, 1] is a positive bounded measurable function and |x| stands for the Euclidean norm of x ∈ R d .We write P for the law of X n , so that for x = y ∈ S: P (X n+1 = y | X n = x) = p(x, y) := ϕ(|y − x|) w S (x) , where we define w S (x) := z∈S: z =x ϕ(|z − x|).Note that the random walk X n is well defined as soon as w S (x) ∈ (0, ∞) for every x ∈ S. In this case, w S = {w S (x) , x ∈ S} is a reversible measure, i.e. w S (x)p(x, y) is symmetric.Since the random walk is irreducible due to the strict positivity of ϕ, w S is the unique invariant measure up to a multiplicative constant.We shall often speak of the random walk (S, ϕ) when we need to emphasize the dependence on the state space S and the function ϕ.Typical special cases of functions ϕ will be the polynomially decaying function ϕ p,α (t) := 1 ∧ t −d−α , α > 0 and the stretched exponential function ϕ e,β (t) := exp(−t β ), β > 0.
We investigate here the transience and recurrence of the random walk X n .We recall that X n is said to be recurrent if for some x ∈ S, the walk started at X 0 = x returns to x infinitely many times with probability one.Because of irreducibility if this happens at some x ∈ S then it must happen at all x ∈ S. X n is said to be transient if it is not recurrent.If we fix S = Z d , we obtain standard homogeneous lattice walks.Transience and recurrence properties of these walks can be obtained by classical harmonic analysis, as extensively discussed e.g. in Spitzer's book [23] (see also Appendix B).For instance, it is well known that for dimension d 3 both (Z d , ϕ e,β ) and (Z d , ϕ p,α ) are transient for all β > 0 and α > 0 while for d = 1, 2, (Z d , ϕ e,β ) is recurrent for all β > 0 and (Z d , ϕ p,α ) is transient iff 0 < α < d.
We shall be interested in the case where S is a locally finite random subset of R d , i.e. the realization of a simple point process on R d .We denote by P the law of the point process.For this model to be well defined for P-almost all S we shall require that, given the choice of ϕ: P (w S (x) ∈ (0, ∞) , for all x ∈ S) = 1 . (1.1) If we look at the set S as a random perturbation of the regular lattice Z d , the first natural question is to find conditions on the law of the point process P and the function ϕ such that (S, ϕ) is P-a.s.transient (recurrent) iff (Z d , ϕ) is transient (recurrent).In this case we say that the random walks (S, ϕ) and (Z d , ϕ) have a.s. the same type.A second question we shall address in this paper is that of establishing almost sure bounds on finite volume effective resistances in the case of certain recurrent random walks of the type (S, ϕ).Before going to a description of our main results we discuss the main examples of point process we have in mind.In what follows we shall use the notation S(Λ) for the number of points of S in any given bounded Borel set Λ ⊂ R d .For any t > 0 and x ∈ R d we write for the cube with side t and the open ball of radius t around x. To check that the models (S, ϕ) are well defined, i.e. (1.1) is satisfied, in all the examples described below the following simple criterion will be sufficient.We write Φ d , for the class of functions ϕ : (0, ∞) → (0, 1] such that ∞ 0 t d−1 ϕ(t)dt < ∞.Suppose the law of the point process P is such that sup Then it is immediate to check that (S, ϕ) satisfies (1.1) for any ϕ ∈ Φ d .
1.1.Examples.The main example we have in mind is the case when P is a homogeneous Poisson point process (PPP) on R d .In this case we shall show that (S, ϕ) and (Z d , ϕ) have a.s. the same type, at least for the standard choices ϕ = ϕ p,α , ϕ e,β .Besides its intrinsic interest as random perturbation of lattice walks we point out that the Poisson point process model arises naturally in statistical physics in the study of the low-temperature conductivity of disordered systems.In this context, the (S, ϕ e,β ) model with β = 1 is a variant of the well known Mott variable-range hopping model, see [13] for more details.The original variable-range hopping model comes with an environment of energy marks on top of the Poisson point process which we neglect here since it does not interfere with the recurrence or transience of the walk.It will be clear that by elementary domination arguments all the results we state for homogeneous PPP actually apply to non-homogeneous PPP with an intensity function which is uniformly bounded from above and away from zero.Motivated by the variable-range hopping problem one could consider point fields obtained from a crystal by dilution and spatial randomization.By crystal we mean any locally finite set Γ ⊂ R d such that for a suitable basis v 1 , v 2 , . . ., v d of R d , one has 3) The spatially randomized and p-diluted crystal is obtained from Γ by first translating Γ by a random vector V chosen with uniform distribution in the elementary cell and then erasing each point with probability 1 − p, independently from the others.One can check that the above construction depends only on Γ and not on the particular G and ∆ chosen.In the case of spatially randomized and p-diluted crystals, P is a stationary point process, i.e. it is invariant w.r.t.spatial translations.It is not hard to check that all the results we state for PPP hold for any of these processes as well for the associated Palm distributions (see [13] for a discussion on the Palm distribution and its relation to Mott variable-range hopping).Therefore, to avoid lengthy repetitions we shall not mention application of our estimates to these cases explicitly in the sequel.We shall also comment on applications of our results to two other classes of point processes: percolation clusters and determinantal point processes.We say that S is a percolation cluster when P is the law of the infinite cluster in super-critical Bernoulli sitepercolation on Z d .For simplicity we shall restrict to site-percolation but nothing changes here if one considers bond-percolation instead.The percolation cluster model has been extensively studied in the case of nearest neighbor walks, see [15,5].In particular, it is well known that the simple random walk on the percolation cluster has almost surely the same type of simple random walk on Z d .Our results will allow to prove that if S is the percolation cluster on Z d then (S, ϕ) has a.s. the same type of (Z d , ϕ), at least for the standard choices ϕ = ϕ p,α , ϕ e,β .
Determinantal point processes (DPP) on the other hand are defined as follows, see [22,4] for recent insightful reviews on DPP.Let K be a locally trace class self-adjoint operator on L 2 (R d , dx).If, in addition, K satisfies 0 K 1 we can speak of the DPP associated to K. Let P, E denote the associated law and expectation.It is always possible to associate a kernel K(x, y) to K such that for any bounded measurable set B ⊂ R d one has where S(B) is the number of points in the set B and 1 B stands for multiplication by the indicator function of the set B, see [22].Moreover, for any family of mutually disjoint subsets where the k-correlation function ρ k satisfies Roughly speaking, these processes are characterized by a tendency towards repulsion between points, and if we consider a stationary DPP, i.e. the case where the kernel satisfies K(x, y) = K(0, y − x), then the repulsive character forces points to be more regularly spaced than in the Poissonian case.A standard example is the sine kernel in d = 1, where K(x, y) = sin(π(x−y)) π(x−y) .Our results will imply for instance that for stationary DPP (S, ϕ) and (Z d , ϕ) have a.s. the same type if ϕ = ϕ p,α (any α > 0) and if ϕ = ϕ e,β with β < d.
1.2.Random resistor networks.Our analysis of the transience and recurrence of the random walk X n will be based on the well known resistor network representation of probabilistic quantities associated to reversible random walks on graphs, an extensive discussion of which is found e.g. in the monographs [8,10].For the moment let us recall a few basic ingredients of the electrical network analogy.We think of (S, ϕ) as an undirected weighted graph with vertex set S and complete edge set {{x, y} , x = y}, every edge {x, y} having weight ϕ(|x − y|).The equivalent electrical network is obtained by connecting each pair of nodes {x, y} by a resistor of magnitude r(x, y) := ϕ(|x − y|) −1 , i.e. by a conductance of magnitude ϕ(|x − y|).We point out that other long-range reversible random walks have already been studied (see for example [2], [3], [17], [19] and references therein), but since the resistor networks associated to these random walks are locally finite and not complete as in our case, the techniques and estimates required here are very different.
One can characterize the transience or recurrence of X n in terms of the associated resistor network.Let {S n } n 1 be an increasing sequence of subsets S n ⊂ S such that S = ∪ n 1 S n and let (S, ϕ) n denote the network obtained by collapsing all sites in S c n = S \ S n into a single site z n (this corresponds to the network where all resistors between nodes in S c n are replaced by infinitely conducting wires but all other wires connecting S n with S n and S n with S c n are left unchanged).For x ∈ S and n large enough such that x ∈ S n , let R n (x) denote the effective resistance between the nodes x and z n in the network (S, ϕ) n .We recall that R n (x) equals the inverse of the effective conductivity C n (x), defined as the current flowing in the network when a unit voltage is applied across the nodes x and z n .On the other hand it is well known that w S (x)R n (x) equals the expected number of visits to x before exiting the set S n for our original random walk (S, ϕ) started at x.The sequence R n (x) is non-decreasing and its limit R(x) is called the effective resistance of the resistor network (S, ϕ) between x and ∞.Then, w S (x)R(x) = lim n→∞ w S (x)R n (x) equals the expected number of visits to x for the walk (S, ϕ) started in x, and the walk (S, ϕ) is recurrent iff R n (x) → ∞ for some (and therefore any) x ∈ S. In light of this, we shall investigate the rate of divergence of R n (x) for specific recurrent models.
Lower bounds on R n (x) can be obtained by the following variational characterization of the effective conductivity C n (x): ( The above infimum is attained when h equals the electrical potential, set to be zero on x and 1 on S c n .From (1.6) one derives Rayleigh's monotonicity principle: the effective conductivity C n (x) decreases whenever ϕ is replaced by ϕ ′ satisfying ϕ ′ (t) ϕ(t) for all t > 0. Upper bounds on R n (x) can be obtained by means of fluxes.We recall that, given a point x ∈ S and a subset B ⊂ S not containing x, a unit flux from x to B is any antisymmetric function f : If B = ∅ then f is said to be a unit flux from x to ∞.The energy E(f ) dissipated by the flux f is defined as To emphasize the dependence on S and ϕ we shall often call E(f ) the (S, ϕ)-energy.
Finally, R n (x), R(x) can be shown to satisfy the following variational principles: In particular, one has the so called Royden-Lyons criterion [21] for reversible random walks: the random walk X n is transient if and only if there exists a unit flux on the resistor network from some point x ∈ S to ∞ having finite energy.An immediate consequence of these facts is the following comparison tool, that we shall often use in the sequel.Lemma 1.1.Let P, P ′ denote two point processes on R d such that P is stochastically dominated by P ′ and let ϕ, ϕ ′ : (0, ∞) → (0, ∞) be such that ϕ Cϕ ′ for some constant C > 0. Suppose further that (1.1) is satisfied for both (S, ϕ) and (S ′ , ϕ ′ ), where S, S ′ denote the random sets distributed according to P and P ′ , respectively.The following holds: (1) if (S, ϕ) is transient P-a.s., then (S ′ , ϕ ′ ) is transient P ′ -a.s.
Proof.The stochastic domination assumption is equivalent to the existence of a coupling of P and P ′ such that, almost surely, S ⊂ S ′ (see e.g.[14] for more details).If (S, ϕ) is transient then there exists a flux f on S with finite (S, ϕ)-energy from some x ∈ S to infinity.We can lift f to a flux on S ′ ⊃ S (from the same x to infinity) by setting it equal to 0 across pairs x, y where either x or y (or both) are not in S.This has finite (S ′ , ϕ)-energy, and since ϕ Cϕ ′ it will have finite (S ′ , ϕ ′ )-energy.This proves (1).for some a, β > 0. Then there exists δ = δ(a, β) > 0 such that (S, ϕ) is a.s.transient whenever ϕ(t) c e −δ t β for some c > 0.
(iii) Set d 1 and suppose that sup Then (S, ϕ) is P-a.s.recurrent whenever (Z d , ϕ 0 ) is recurrent, where ϕ 0 is given by The proof of these general statements is given is Section 2. It relies on rather elementary arguments not far from the rough embedding method described in [10, Chapter 2].In particular, to prove (i) and (ii) we shall construct a flux on S from a point x ∈ S to infinity and show that it has finite (S, ϕ)-energy under suitable assumptions.The flux will be constructed using comparison with suitable long-range random walks Z d .Point (iii) of Theorem 1.2 is obtained by exhibiting a candidate for the electric potential in the network (S, ϕ) which produces a vanishing conductivity.Again the construction is achieved using comparison with long-range random walks on Z d .
Despite the simplicity of the argument, Theorem 1.2 already captures non-trivial facts such as e.g. the transience of the super-critical percolation cluster in dimension two with ϕ = ϕ p,α , α < 2.More generally, combining (i) and (iii) of Theorem 1.2 we shall obtain the following corollary.
Corollary 1.3.Fix d 1.Let P be one of the following point processes: a homogeneous PPP; the infinite cluster in super-critical Bernoulli site-percolation on Z d ; a stationary DPP on R d .Then (S, ϕ p,α ) has a.s. the same type as (Z d , ϕ p,α ), for all α > 0.
We note that for the transience results (i) and (ii) we only need to check the sufficient conditions (1.12) and (1.13) on the function ψ(t).Remarks on how to prove bounds on ψ(t) for various processes are given in Subsection 2.2.Conditions (1.12) and (1.13) in Theorem 1.2 are in general far from optimal.We shall give a bound that improves point (i) in the case d = 1, see Proposition 1.7 below.
The limitations of Theorem 1.2 become more important when ϕ is rapidly decaying and d 3.For instance, if P is the law of the infinite percolation cluster, then ψ(t) satisfies a bound of the form e −c t d−1 , see Lemma 2.5 below.Thus in this case point (ii) would only allow to conclude that there exists a = a(p) > 0 such that, in d 3, (S, ϕ) is P-a.s.transient if ϕ(t) Ce −a t d−1 .However, the well known Grimmett-Kesten-Zhang theorem about the transience of nearest neighbor random walk on the infinite cluster in d 3 ( [15], see also [5] for an alternative proof) together with Lemma 1.1 immediately implies that (S, ϕ) is a.s.transient for any ϕ ∈ Φ d .Similarly, one can use stochastic domination arguments to improve point (ii) in Theorem 1.2 for other processes.To this end we say that the process P dominates (after coarse-graining) super-critical Bernoulli site-percolation if P is such that for some L ∈ N the random field stochastically dominates the i.i.d.Bernoulli field on Z d with some super-critical parameter p.Here χ(•) stands for the indicator function of an event.In particular, it is easily checked that any homogeneous PPP dominates super-critical Bernoulli site-percolation.
For DPP defined on Z d stochastic domination w.r.t.Bernoulli can be obtained under suitable hypothesis on the kernel K, see [11].We are not aware of analogous conditions in the continuum that would imply that DPP dominates super-critical Bernoulli sitepercolation.In the latter cases we have to content ourselves with point (ii) of Theorem 1.2 (which implies point 3 in Corollary 1.4 below).We summarize our conclusions for ϕ = ϕ e,β in the following Corollary 1.4. 1.Let P be any of the processes considered in Corollary 1.3.Then (S, ϕ e,β ) is a.s.recurrent in d = 1, 2, for any β > 0.
2. Let P be the law of the infinite cluster in super-critical Bernoulli site-percolation on Z d or a homogeneous PPP or any other process that dominates super-critical Bernoulli site-percolation.Then (S, ϕ e,β ) is a.s.transient in d 3, for any β > 0.
We point our that, by the same proof, point 2) above remains true if (S, ϕ e,β ) is replaced by (S, ϕ), ϕ ∈ Φ d .
1.4.Bounds on finite volume effective resistances.When a network (S, ϕ) is recurrent the effective resistances R n (x) associated to the finite sets S n := S ∩ [−n, n] d diverge, see (1.8), and we may be interested in obtaining quantitative information on their growth with n.We shall consider in particular the case of point processes in dimension d = 1, with ϕ = ϕ p,α , α ∈ [1, ∞), and the case d = 2 with ϕ = ϕ p,α , α ∈ [2, ∞).By Rayleigh's monotonicity principle, the bounds given below apply also to (S, ϕ), whenever ϕ Cϕ p,α .In particular, they cover the stretched exponential case (S, ϕ e,β ).
We say that the point process P is dominated by an i.i.d.field if the following condition holds: There exists L ∈ N such that the random field For the results in d = 1 we shall require the following exponential moment condition on the dominating field Γ: There exists ε > 0 such that (1.17) For the results in d = 2 it will be sufficient to require the existence of the fourth moment: It is immediate to check that any homogeneous PPP is dominated by an i.i.d.field in the sense described above and the dominating field Γ satisfies (1.17).Moreover, this continues to hold for non-homogeneous Poisson process with a uniformly bounded intensity function.We refer the reader to [11,14] for examples of determinantal processes satisfying this domination property.
Theorem 1.5.Set d = 1, ϕ = ϕ p,α and α 1. Suppose that the point process P is dominated by an i.i.d.field satisfying (1.17).Then, for P-a.a.S the network (S, ϕ) satisfies: given x ∈ S there exists a constant c > 0 such that for all n 2 such that x ∈ S n .
Theorem 1.6.Set d = 2, ϕ = ϕ p,α and α 2. Suppose that P is dominated by an i.i.d.field satisfying (1.18).Then, for P-a.a.S the network (S, ϕ) satisfies: given x ∈ S there exists a constant c > 0 such that for all n 2 such that x ∈ S n .
The proof of Theorem 1.5 and Theorem 1.6 is given in Section 3. The first step is to reduce the network (S, ϕ) to a simpler network by using the domination assumption.In the proof of Theorem 1.5 the effective resistance of this simpler network is then estimated using the variational principle (1.6) with suitable trial functions.
In the proof of Theorem 1.6 we are going to exploit a further reduction of the network which ultimately leads to a one-dimensional nearest neighbor network where effective resistances are easier to estimate.This construction uses an idea already appeared in [18], see also [7] and [1] for recent applications, which allows to go from long-range to nearest neighbor networks, see Section 3 for the details.Theorem 1.6 could be also proved using the variational principle (1.6) for suitable choices of the trial function, see the remarks in Section 3.
It is worthy of note that the proofs of these results are constructive in the sense that they do not rely on results already known for the corresponding (Z d , ϕ p,α ) network.In particular, the method can be used to obtain quantitative lower bounds on R n (x) for the deterministic case S ≡ Z d , which is indeed a special case of the theorems.In the latter case the lower bounds obtained here as well suitable upper bounds are probably well known but we were not able to find references to that in the literature.In appendix B, we show how to bound from above the effective resistance R n (x) of the network (Z d , ϕ p,α ) by means of harmonic analysis.The resulting upper bounds match with the lower bounds of Theorems 1.5 and 1.6, with exception of the case d = 1, α = 2 where our upper and lower bounds differ by a factor √ log n.
1.5.Constructive proofs of transience.While the transience criteria summarized in Corollary 1.3 and Corollary 1.4 are based on known results for the deterministic networks (Z d , ϕ) obtained by classical harmonic analysis, it is possible to give constructive proofs of these results by exhibiting explicit fluxes with finite energy on the network under consideration.We discuss two results here in this direction.The first gives an improvement over the criterium in Theorem 1.2, part (i), in the case d = 1.This can be used in particular to give a "flux-proof" of the well known fact that (Z, ϕ p,α ) is transient for α < 1.The second result gives a constructive proof of transience of a deterministic network, which, in turn, reasoning as in the proof of Theorem 1.2 part (i), gives a flux-proof that (Z 2 , ϕ p,α ) is transient for α < 2.
In order to state the one-dimensional result, it is convenient to number the points of S as S = {x i } i∈I where x i < x i+1 , x −1 < 0 x 0 and N ⊂ I or −N ⊂ I (we assume that |S| = ∞, P-a.s., since otherwise the network is recurrent).For simplicity of notation we assume below that N ⊂ I, P-a.s.The following result can be easily extended to the general case by considering separately the conditional probabilities P(•|N ⊂ I) and P(•|N ⊂ I), and applying a symmetry argument in the second case.Proposition 1.7.Take d = 1 and α ∈ (0, 1).Suppose that for some positive constants c, C it holds Then P-a.s.(S, ϕ) is transient.In particular, if P is a renewal point process such that then P-a.s.(S, ϕ) is transient.
Suppose that P is a renewal point process and write Then (1.23) certainly holds as soon as e.g.ψ satisfies ψ(t) Ct −(1+α+ε) for some positive constants C, ε.We can check that this improves substantially over the requirement in Theorem 1.2, part (i), since if ψ is defined by (1.10), then we have, for all t > 1: The next result concerns the deterministic two-dimensional network (S * , ϕ p,α ) defined as follows.Identify R 2 with the complex plane C, and define the set S * := ∪ ∞ n=0 C n , where Theorem 1.8.The network (S * , ϕ p,α ) is transient for all α ∈ (0, 2).
This theorem, together with the comparison techniques developed in the next section (see Lemma 2.1 below), allows to recover by a flux-proof the transience of (Z 2 , ϕ p,α ) for α ∈ (0, 2).The proofs of Proposition 1.7 and Theorem 1.8 are given in Section 4.1.

Recurrence and transience by comparison methods
Let S 0 be a given locally finite subset of R d and let (S 0 , ϕ 0 ) be the associated random walk.We assume that w S 0 (x) < ∞ for all x ∈ S 0 and that ϕ 0 (t) > 0 for all t > 0. Recall that in the resistor network picture every node {x, y} is given the resistance r 0 (x, y) := ϕ 0 (|x − y|) −1 .To fix ideas we may think of S 0 = Z d and either ϕ 0 = ϕ p,α or ϕ 0 = ϕ e,β .(S 0 , ϕ 0 ) will play the role of the deterministic background network.
Let P denote a simple point process on R d , i.e. a probability measure on the set Ω of locally finite subsets S of R d , endowed with the σ-algebra F generated by the counting maps N Λ : Ω → N ∪ {0}, where N Λ (S) = S(Λ) is the number of points of S that belong to Λ and Λ is a bounded Borel subset of R d .We shall use S to denote a generic random configuration of points distributed according to P. We assume that P and ϕ are such that (1.1) holds.
Next, we introduce a map φ : S 0 → S, from our reference set S 0 to the random set S. For any x ∈ S 0 we write φ(x) = φ(S, x) for the point in S which is closest to x in Euclidean distance.If the Euclidean distance from x to S is minimized by more than one point in S then choose φ(x) to be the one with lowest lexicographic order.This defines a measurable map Ω ∋ S → φ(S, x) ∈ R d for every x ∈ S 0 .
For any point u ∈ S define the cell By construction {V u , u ∈ S} determines a partition of the original vertex set S 0 .Clearly, some of the V u may be empty, while some may be large (if S has large "holes" with respect to S 0 ).Let N (u) denote the number of points (of S 0 ) in the cell V u .We denote by E the expectation with respect to P.
Proof.Without loss of generality we shall assume that 0 ∈ S 0 .Since (S 0 , ϕ 0 ) is transient, from the Royden-Lyons criterion recalled in Subsection 1.2, we know that there exists a unit flux f : S 0 × S 0 → R from 0 ∈ S 0 to ∞ with finite (S 0 , ϕ 0 )-energy.By the same criterion, in order to prove the transience of (S, ϕ) we only need to exhibit a unit flux from some point of S to ∞ with finite (S, ϕ)-energy.To this end, for any u, v ∈ S we define If either V u or V v are empty we set θ(u, v) = 0. Note that the above sum is finite for all u, v ∈ S, P-a.s.Indeed condition (2.1) implies that N (φ(x)) < ∞ for all x ∈ S 0 , P-a.s.Thus, θ defines a unit flux from φ(0) to infinity on (S, ϕ).Indeed, for every u, v ∈ S we have θ(u, v) = −θ(v, u) and for every u = φ(0) we have v∈S θ(u, v) = 0.Moreover, The energy of the flux θ is given by It follows that Since f has finite energy on (S 0 , ϕ 0 ) we see that condition (2.1) implies E[E(θ)] < ∞.In particular, this shows that P-a.s.there exists a unit flux θ from some point u 0 ∈ S to ∞ with finite (S, ϕ)-energy.
To produce an analogue of Lemma 2.1 in the recurrent case we introduce the set S = S ∪S 0 and consider the network ( S, ϕ).From monotonicity of resistor networks, recurrence of (S, ϕ) is implied by recurrence of ( S, ϕ).We define the map φ ′ : S → S 0 , from S to the reference set S 0 as the map φ introduced before, only with S 0 replaced by S and S replaced by S 0 .Namely, given x ∈ S we define φ ′ (x) as the point in S 0 which is closest to x in the Euclidean distance (when there is more than one minimizing point, we take the one with lowest lexicographic order).Similarly, for any point x ∈ S 0 we define x , x ∈ S 0 } determines a partition of S. Note that in this case all V ′ x are non-empty (V ′ x contains x ∈ S).
As an example, if 2 ) for any x ∈ Z d .Lemma 2.2.Suppose that (S 0 , ϕ 0 ) is recurrent and that P-a.s.V ′ x is finite for all x ∈ S 0 .If there exists C < ∞ such that for all x = y in S 0 it holds then (S, ϕ) is P-a.s.recurrent.
Proof.Without loss of generality we shall assume that 0 ∈ S 0 .Set S 0,n = S 0 ∩ [−n, n] d , collapse all sites in S c 0,n = S 0 \ S 0,n into a single site z n and call c(S 0,n ) the effective conductivity between 0 and z n , i.e. the net current flowing in the network when a unit voltage is applied across 0 and z n .Since (S 0 , ϕ 0 ) is recurrent we know that c(S 0,n ) → 0, n → ∞.
Recall that c(S 0,n ) satisfies where ψ n is the electric potential, i.e. the unique function on S 0 that is harmonic in S 0,n , takes the value 1 at 0 and vanishes out of S 0,n .Given S ∈ Ω, set Note that S n is an increasing sequence of finite sets, covering all S. Collapse all sites in S c n into a single site z n and call c( S n ) the effective conductivity between 0 and z n (by construction 0 ∈ S n ).From the Dirichlet principle (1.6) we have for any g : S → [0, 1] such that g(0) = 1 and g = 0 on S c n .Choosing g(u) = ψ n (φ ′ (u)) we obtain c( S n ) 1 2 x,y∈S 0 From the assumption (2.4) and the recurrence of (S 0 , ϕ 0 ) implying that (2.5) goes to zero, we deduce that ) is monotone decreasing we deduce that c( S n ) → 0, P-a.s.This implies P-a.s.recurrence of ( S, ϕ) and the claim follows.
2.1.Proof of Theorem 1.2.We first prove part (i) of the theorem, by applying the general statement derived in Lemma 2.1 in the case S 0 = Z d and ϕ 0 = ϕ p,α .Since (S 0 , ϕ p,α ) is transient whenever d 3, or d = 1, 2 and 0 < α < d, we only need to verify condition (2.1).For the moment we only suppose that ψ(t) C ′ t −γ for some γ > 0. Let us fix p, q > 1 s.t.1/p + 1/q = 1.Using Hölder's inequality and then Schwarz' inequality, we obtain (2.6) From (2.7) and the fact that |x − y| 1 we derive that Now we observe that |φ(z) − z| t if and only if B z,t ∩ S = ∅.Hence we can estimate whenever γ > p(d + α) as we assume.Therefore, using |x − y| 1, from (2.8) we see that for any x = y in Z d : (2.9) Next, we estimate the expectation E N (φ(x)) 2q from above, uniformly in x ∈ Z d .To this end we shall need the following simple geometric lemma.
Lemma 2.3.Let E(x, t) be the event that S ∩ B(x, t) = ∅ and S ∩ B(x ± 3 √ d te i , t) = ∅, where {e i : 1 i d} is the canonical basis of R d .Then, on the event E(x, t) we have φ(x) ∈ B(x, t), i.e. |φ(x) − x| < t, and Assuming for a moment the validity of Lemma 2.3 the proof continues as follows.From Lemma 2.3 we see that, for a suitable constant c 5 , the event N (φ(x)) > c 5 t d implies that at least one of the 2d + 1 balls B(x, t), B(x ± 3 √ dte i , t) must have empty intersection with S. Since B(x ± ⌊3 √ dt⌋e i , t − 1) ⊂ B(x ± 3 √ dte i , t) for t 1, we conclude that Taking c 6 such that c as soon as γ > 2qd.
Due to (2.6), (2.9) and (2.10), the hypothesis (2.1) of Lemma 2.1 is satisfied if ψ(t) C t −γ for all t 1, where γ is a constant satisfying We observe that the functions (1, ∞) ∋ p → p(d+α) and (1, ∞) ∋ p → 2pd p−1 are respectively increasing and decreasing and intersect in only one point p * = 3d+α d+α .Hence, optimizing over p, it is enough to require that This concludes the proof of Theorem 1.2 (i).
In order to prove the second one we proceed as follows.(2.11) we are sure that the distance from z of each point in S ∩ B(0, t) is larger than the distance from z of each point of S ∩ B(mte 1 , t).Hence it cannot be that z ∈ V φ(0) .In order to prove (2.11), we first observe that the map (0, By the mean value theorem applied to the function This completes the proof of (2.11).
For any β > 0 we can find a constant Therefore, using Schwarz' inequality we have where, using (1.13), the last integral is finite for γ < a. Taking γ = δ c 2 and δ sufficiently small we arrive at the conclusion that uniformly in x, Clearly, r 0 (x, y) defines a transient resistor network on Z d and the same claim for r 0 (x, y) follows from monotonicity.This ends the proof of Part (ii) of Theorem 1.2.
Proof of Theorem 1.2 Part (iii).Here we use the criterion given in Lemma 2.2 with S 0 = Z d .With this choice of S 0 we have that Using the Schwarz' inequality and condition (1.14) the last expression is bounded by c 2 ϕ 0 (x, y).This implies condition (2.4) and therefore the a.s.recurrence of (S, ϕ).

Proof of
where λ i (B) denote the eigenvalues of K1 B for any bounded Borel set B ⊂ R d .In particular, condition (1.12) is satisfied if for some constants C > 0 and γ > 3d + α.If P is a stationary DPP then ψ(t) e −δ t d , t > 0, for some δ > 0.
Finally, condition (1.14) reads In particular, condition (1.14) always holds if P is a stationary DPP.
Proof.It is known (see [4] and [22]) that for each bounded Borel set B ⊂ R d the number of points S(B) has the same law of the sum i B i , B i 's being independent Bernoulli random variables with parameters λ i (B).This implies identity (2.13).Since 1 − x e −x , x 0, we can bound the r.h.s. of (1.10) by i e −λ i (B) = e −T r(K1 B ) .This identity and (1.4) imply (2.14).If the DPP is stationary then K(u, u) ≡ K(0) > 0 and therefore ψ(t) e −K(0) t d .Finally, (2.15) follows from the identity E[S(Q(x, 1)) 2 ] = i λ i (Q(x, 1))+ i j: j =i λ i (Q(x, 1))λ j (Q(x, 1)), again a consequence of the fact that S(Q(x, 1)) is the sum of independent Bernoulli random variables with parameters λ i (Q(x, 1)).Since the sum of the λ i (Q(x, 1))'s is finite, E[S(Q(x, 1)) 2 ] < ∞ for any x ∈ Z d .If the DPP is stationary it is uniformly finite.
The next lemma allows to estimate ψ(t) in the case of percolation clusters.Lemma 2.5.Let P be the law of the infinite cluster in super-critical Bernoulli site (or bond) percolation in Z d , d 2. Then there exist constants k, δ > 0 such that Proof.The lower bound follows easily by considering the event that e.g. the cube centered at the origin with side n/2 has all the boundary sites (or bonds) unoccupied.To prove the upper bound one can proceed as follows.Let K n (γ), γ > 0, denote the event that there exists an open cluster C inside the box Known estimates (see e.g.Lemma (11.22) in Grimmett's book [16] for the case d = 2 and Theorem 1.2 of Pisztora's [20] for d 3) imply that there exist constants k 1 , δ 1 , γ > 0 such that (2.16) On the other hand, let C x denote the open cluster at x ∈ Z d and write C ∞ for the infinite open cluster.From [16,Theorem (8.65)] we have that there exist constants k 2 , δ 2 such that for any x ∈ Z d and for any γ > 0: (2.17) Now we can combine (2.16) and (2.17) to prove the desired estimate.For any n we write The last term in this expression is bounded using (2.16).The first term is bounded by Using (2.17) we arrive at We are now ready to finish the proof of Corollary 1.3.It is clear from the previous lemmas that in all cases we have both conditions (1.12) and (1.14).Moreover it is easily verified that (Z d , ϕ 0 ) and (Z d , ϕ p,α ) have the same type, when ϕ 0 is defined by (1.15) with ϕ = ϕ p,α .This ends the proof of Corollary 1.3.

Proof of Corollary 1.4.
It is easily verified that (Z d , ϕ 0 ) and (Z d , ϕ e,β ) have the same type, when ϕ 0 is defined by (1.15) with ϕ = ϕ e,β .Therefore the statement about recurrency follows immediately from Theorem 1.2, Part (iii) and the fact that in all cases (1.14) is satisfied (see the previous Subsection).
To prove the second statement we recall that our domination assumption and Strassen's theorem imply that on a suitable probability space (Ω, P) one can define the random field (σ 1 , σ 2 ) ∈ {0, 1} Z d × {0, 1} Z d such that σ 1 has the same law of the infinite cluster in a super-critical Bernoulli site-percolation on Z d , σ 2 has the same law of the random field σ defined in (1.16) and σ 1 σ 2 , P-a.s., i.e. σ 1 (x) σ 2 (x) for all x ∈ Z d , P-a.s.To each σ 1 we associate the nearest-neighbor resistor network N 1 with nodes {x ∈ Z d : σ 1 (x) = 1} such that nearest-neighbor nodes are connected by a conductance of value c 0 = c 0 (L) > 0 to be determined below.From the result of [15] we know that N 1 is transient a.s.Now, for each cube Q Lx,L intersecting S we fix a point x ∈ Q Lx,L ∩ S (say the one with least lexicographic order).If we keep all points x's belonging to the infinite cluster in σ 2 and neglect all other points of S we obtain a subnetwork ( S, ϕ) of (S, ϕ).If c 0 is sufficiently small we have ϕ(y, z) c 0 for all y, z ∈ S such that y Reasoning as in the proof of Lemma 1.1 then immediately implies the a.s.transience of (S, ϕ).Note that this actually works for any ϕ ∈ Φ d .
To prove the third statement we observe that for stationary DPP ψ(t) e −δ t d , see Lemma 2.4.Therefore the claim follows from Theorem 1.2, Part (ii).

Lower bounds on the effective resistance
Assume that P is dominated by an i.i.d.field Γ as stated before Theorem 1.5 and suppose the domination property holds with some fixed L ∈ N. We shall write Q v for the cube Q vL,L .To prove Theorem 1.5 and Theorem 1.6 we only need to show that, given v 0 ∈ Z d , for P-a.a.S there exists a positive constant c such that for all x ∈ S ∩ Q v 0 L,L the lower bounds (1.19) and (1.20) on R n (x) hold.We restrict to v 0 = 0, since the general case can be treated similarly.
We start by making a first reduction of the network which uses the stochastic domination assumption.This procedure works in any dimension d.First, we note that it is sufficient to prove the bounds in the two theorems for the quantity Rn (x), defined as the effective resistance from x to Q c 0,2Ln , instead of R n (x) which is the effective resistance from x to Q c 0,2n .In particular, there is no loss of generality in taking L = 1, in which case Rn (x) = R n (x).
The next observation is that, by monotonicity, R n (x) is larger than the same quantity computed in the network obtained by collapsing in a single node v all points in each cube Q v , v ∈ Z d .We now have a network with nodes on the points of Z d (although some of them may be empty).Note that across two nodes u, v we have N u N v wires each with a resistance bounded from below by ρ u,v := c |u − v| d+α for a suitable (non-random) constant c > 0.Moreover, using the stochastic domination assumption we know that N u Γ u for all u ∈ Z d , and we can further lower the resistance by considering the network where each pair of nodes u, v is connected by Γ u Γ v wires each with the resistance ρ u,v .Moreover, we can further lower the resistance by adding a point to the origin.Hence, from now on, we understand that Γ u is replaced by Γ u + 1 if u = 0. We call (Γ, ρ) this new network.Thus the results will follow once we prove that for (Γ, ρ) the effective resistance from 0 to Q c 0,2n = {u ∈ Z d : u ∞ > n} satisfies the desired bounds.From now on we consider the cases d = 1 and d = 2 separately.
3.1.Proof of Theorem 1.5.Set d = 1.We further reduce the network (Γ, ρ) introduced above by collapsing in a single node v each pair {v, −v}.This gives a network on {0, 1, 2, . . .} where across each pair 0 i < j there are now Γ i Γ j wires, where Γ i := Γ i +Γ −i (i = 0) and Γ 0 := Γ 0 (recall that by Γ 0 we now mean the original Γ 0 plus 1).Each of these wires has a resistance at least ρ i,j and thus we further reduce the network by assigning each wire the same resistance ρ i,j .We shall call ( Γ, ρ) this new network and R n (0) its effective resistance from 0 to Q c 0,2n .An application of the variational formula (1.6) to the network ( Γ, ρ) yields the upper bound for any sequence {f i } i 0 such that f i = f (i), f being a non-decreasing function on [0, ∞) taking value 0 only at the origin.
Next, we choose f as Note that f satisfies the differential equation Moreover, f is increasing on [0, ∞), f (0) = 0 and f k = f (k) behaves as Here . Since g α is non-increasing we have the concavity bounds Let us first prove the theorem for the easier case α > 2. We point out that here we do not need condition (1.17) and a finite first moment condition suffices.Indeed, set ξ i := j>i (j − i) 1−α Γ j .This random variables are identically distributed and have finite first moment since α > 2. Note that Γ i and ξ i are independent so that E[Γ i ξ i ] < ∞.From the ergodic theorem it follows that there exists a constant C such that P-a.s.
for all n sufficiently large.Due to (3.1), we conclude that C n (0) n −2 n i=0 Γ i ξ i C n −1 and the desired bound R n (0) c n follows.
The case 1 α 2 requires more work.Thanks to our choice of f , we shall prove the following deterministic estimate.Lemma 3.1.There exists a constant C < ∞ such that for any α 1 Let us assume the validity of Lemma 3.1 for the moment and define the random variables Let us show that P-a.s.ξ i satisfies the same bound as From assumption (1.17) we know Λ(λ) < ∞ for all λ ε for some ε > 0.Moreover, Λ(λ) is convex and Λ(λ) c λ for some constant c, for all λ ε.Therefore, using Lemma 3.1 we have, for some new constant C: provided the numbers a i > 0 satisfy a i (j − i) −1−α (f j − f i ) 2 ε for all j > i > 0. Note that the last requirement is satisfied by the choice a i := ε/g α (i) 2 since, using (3.5): for j > i, α 1.This will be our choice of a i for 1 α 2. From (3.7) we have 2 and i large enough.Therefore, if c 1 is sufficiently large, the left hand side in (3.8) is summable in i ∈ N and the Borel Cantelli lemma implies that P-a.s.we have ξ i c 2 g α (i), c 2 := 2c 1 ε −1 , for all i i 0 , where i 0 is an a.s.finite random number.
Next, we write The first term is an a.s.finite random number.The second term is estimated as follows.
First, note that since by concavity Then we estimate If i is large enough (so that g α (i) ε) we can estimate Λ(g α (i)) c g α (i).Using (3.10) we then have, for c 3 large enough Since f n log n for all α 1 we see that, if c 3 is sufficiently large, the Borel Cantelli lemma implies that the second term in (3.9) is P-a.s.bounded by 2c 3 f n for all n n 0 for some a.s.finite random number n 0 .It follows that there exists an a.s.positive constant c > 0 such that R n (0) c f n .The proof of Theorem 1.5 is thus complete once we prove the deterministic estimate in Lemma 3.1.
Proof of Lemma 3.1.We only need to consider the cases α ∈ [1,2].We divide the sum in two terms We can estimate the first term by using the concavity of f and equation (3.3): (3.12) As far as the second term is concerned, first observe that (f j − f i )/(j − i) is non-increasing in j (by concavity of f ) and so is the general term of the series.As a consequence j>2i In the case α = 1 we get, for any i 1, In the case α > 1 we have, so that, for 1 < α < 2, there are two positive constants C and and, for α = 2, there are two positive constants C and 3.2.Proof of Theorem 1.6.To prove Theorem 1.6 we shall make a series of network reductions which allow us to arrive at a nearest neighbor one-dimensional problem.We start from the network (Γ, ρ) defined at the beginning of this section.We write The next reduction is obtained by collapsing all nodes u ∈ F a into a single node for each a ∈ N.
Once all nodes in each F a are identified we are left with a one-dimensional network with nodes a ∈ {0, 1, . . .}. Between nodes a and b we have a total of u∈Fa Γ u v∈F b Γ v wires, with a wire of resistance ρ u,v for each u ∈ F a and v ∈ F b .Finally, we perform a last reduction which brings us to a nearest-neighbor one-dimensional network.To this end we consider a single wire with resistance ρ u,v between node a and node b, with a < b − 1.This wire is equivalent to a series of (b − a) wires, each with resistance ρ u,v /(b − a).That is we can add (b − a − 1) fictitious points to our network in such a way that the effective resistance does not change.Moreover the effective resistance decreases if each added point in the series is attached to its corresponding node a + i, i = 1, . . ., b − a − 1, in the network.If we repeat this procedure for each wire across every pair of nodes a < b − 1 then we obtain a nearest neighbor network where there are infinitely many wires in parallel across any two consecutive nodes.In this new network, across the pair i−1, i we have a resistance R i−1,i such that Moreover, the reductions described above show that Therefore Theorem 1.6 now follows from the estimates on R i−1,i given in the next lemma.
Lemma 3.2.There exists a positive constant c such that P-almost surely, for i sufficiently large Proof.We first show that E(φ i ) Cω i , where where E denotes expectation w.r.t. the field {Γ u , u ∈ Z 2 }.Thanks to Lemma A.1 given in the Appendix, from (3.18) we have Next we estimate b i (b − a) −α c 2 (i − a) 1−α , so that using the Riemann integral we obtain Hence, for C large we can estimate where we use P to denote the law of the variables {Γ u }.
The proof then follows from the Borel-Cantelli Lemma and the following estimate to be established below: There exists C < ∞ such that for all i ∈ N To prove (3.22) we write where the sums are over a = (a 1 , . . ., a 4 ), b = (b 1 , . . ., b 4 ) such that a k < i b k , u ∼ a stands for the set of u = (u 1 , . . ., u 4 ) such that u k ∈ F a k , and we have defined, for u ∼ a, v ∼ b: From the independence assumption on the field {Γ u } we know that G(u, v) = 0 unless for every k = 1, . . ., 4 there exists a k ′ = 1, . . ., Moreover, when this condition is satisfied using (1.18) we can easily bound G(u, v) C for some constant C. By symmetry we may then estimate We claim that each of the three terms in the summation above is of order i 2 as i grows.This will prove the desired estimate (3.22).
The first term in (3.24) satisfies where Similarly the third term in (3.24) is estimated by B(i) 2 , with Finally, the middle term in (3.24) is estimated by the product A(i)B(i).Therefore, to prove (3.22) it suffices to show that A(i) C i and B(i) C i. Using Lemma A.1 we see that

This bound yields
for some new constant C, where we have used the fact that Using the Riemann integral we obtain This proves that A(i) = O(i).Similarly, from Lemma A.1 we see that Therefore where we have used the estimate and the fact that for α 2 we have We remark that a proof of Theorem 1.6 could be obtained by application of the variational principle (1.6) as in the proof of Theorem 1.5.To see this one can start from the network (Γ, ρ) introduced at the beginning of this section and choose a trial function that is constant in each F a .Then, for any non-decreasing sequence (f 0 , f 1 , . . . ) such that f 0 = 0 and f k > 0 eventually, one has R n (x) A n (f ) where (3.26) We then choose f k = log(1 + k) for α > 2 and f k = log(log(e + k)) for α = 2 and the desired conclusions will follow from suitable control of the fluctuations of the random sum appearing in (3.26).Here the analysis is slightly more involved than that in the proof of Theorem 1.5 and it requires estimates as in (3.24) above.Moreover, one needs a fifth moment assumption with this approach instead of the fourth moment condition (1.18).Under this assumption, and using Lemma A.1, it is possible to show that a.s.there exists a constant c such that (c(δ) being the normalizing constant 1/ k 1 k −δ ) and define recursively the sequence

.2)
Proof.Let {X i } i 1 be a family of IID random variables with P ( In particular, if 1 < δ < 2 we can use Theorem B of [9] and get (4.2) with u(n) instead f (n), where u(n) is defined as follows: Consider the random walk S n on the set N ∪ {0}, starting at 0, S 0 = 0, and defined as Trivially u(0) = 1, while the Markov property of the random walk S n gives for n 1 that Hence, f (n) and u(n) satisfy the same system of recursive identities and coincide for n = 0, thus implying that f (n) = u(n) for each n ∈ N.
We have now all the tools in order to prove Proposition 1.7: Proof of Proposition 1.7.We shall exhibit a finite energy unit flux f (•, •) from x 0 to infinity in the network (S, ϕ).To this end we define f (•, •) as follows where f (m), q m are defined as in the previous lemma for some δ ∈ (1, 2) that will be fixed below.Since ϕ Cϕ p,α , the energy where r p,α (x, y) := 1/ϕ p,α (|x − y|).Hence, due to the previous lemma we obtain that In order to prove that the energy E(f ) is finite P-a.s., it is enough to show that E(E(f )) is finite for some δ ∈ (1, 2).To this end we observe that, due to assumption (1.22) and since r p,α (x k , for suitable constants c 1 , c 2 .Hence, the mean energy is finite if 2δ−4 < −1 and 1+α−2δ < −1.In particular for each α ∈ (0, 1) one can fix δ ∈ (1, 2) satisfying the above conditions.This concludes the proof of the transience of (S, ϕ) for P-a.a. S. It remains to verify assumption (1.22) whenever P is a renewal point process such that E(( To this end we observe that by convexity Since by the renewal property (x k+1 − x k ) k 0 is a sequence of i.i.d.random variables, the mean of the last expression equals u 1+α E((x 1 − x 0 ) 1+α ) = cu 1+α .Therefore, (1.22) is satisfied.
4.2.Proof of Theorem 1.8.We recall that S * = ∪ n 0 C n , where and C is identified with R 2 .In order to introduce more symmetries we consider a family of "rotations" of the C n 's: given θ = (θ n ) n 0 a sequence of independent random variables with uniform law on (− π n+1 , + π n+1 ) we define and for x in C n we use the notation We will construct a unit flow f θ from 0 to infinity on S θ * := ∪ n C θ n and will make an average over θ to build a new flow f on S. In order to describe the flow f θ , we consider the probability kernel q k = c(δ)k −δ , δ ∈ (1, 2), introduced in Lemma 4.1.The value of δ will be chosen at the end.We build f θ driving a fraction q n−m of the total flow arriving in a site x θ ∈ C θ m to each C θ n with n > m, in such a way that for each site y ∈ C θ n the flow received from x θ is proportional to ϕ p,α (x θ , y θ ).We have then, for all n > m, x ∈ C m and y and f θ (•) defined recursively as Note that the quantity is independent from θ and it is defined recursively by By Lemma 4.1 and the condition δ ∈ (1, 2) we have We can now define our flow f on (S * , ϕ p,α ).For all m < n, x ∈ C m and y ∈ C n we set where the expectation is w.r.t.θ.Taking the conditional expectation in (4.9) we get By radial symmetry the last factor does not depend neither on x nor on θ m and taking the conditional expectation in (4.12) we get where θ is a unit vector (the integral does not depend on the choice of θ).This integral is positive and finite for α ∈ (0, 2) and (B.3) follows.Using (B.3) the integrability criterion (B.2) implies that for d = 1 the RW is transient if and only if α ∈ (0, 1), while for d = 2 the RW is transient for any α ∈ (0, 2).The only case remaining is d = 2, α = 2.This apparently is not covered explicitly in [23].However, one can modify the argument above to obtain that for any d 1, α = 2: lim x 2  1 dx |x| d+2 + O 1/ log(ε −1 ) , where x 1 is the first coordinate of the vector x = (x 1 , . . ., x d ).The integral appearing in the first term above is, apart from a constant factor, δ ε r −1 dr = log(ε −1 ) + const.This proves the claim (B.5).B.2. Effective resistance estimates.Let R n := R n (0) be the effective resistance associated to the box {x ∈ Z d , x ∞ n}.As already discussed in the introduction, 1 c R n (where c > 0 is the constant in (B.1)) equals the expected number of visits to the origin before visiting the set {x ∈ Z d , x ∞ > n} for the random walk X with X 0 = 0. We are going to give upper bounds on R n in the recurrent cases d = 1, 2, α min{d, 2}.By comparison with the simple nearest neighbor random walk we have that (for any α) R n C log n if d = 2 and R n C n if d = 1.Due to Theorems 1.5 and 1.6, this estimate is of the correct order whenever p(x) has finite second moment (α > 2).The remaining cases are treated as follows.
We claim that for some constant C Taking C large enough and using γ n = O(n −α ) shows that (B.9) holds.This ends the proof of (B.6).
.27) Once this estimate is available the proof follows from simple calculations.4. Proof of Proposition 1.7 and Theorem 1.8 4.1.Proof of Proposition 1.7.The proof of Proposition 1.7 is based on the following technical lemma related to renewal theory: Lemma 4.1.Given δ > 1, define the probability kernel