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El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y} Vol. 14 (2009), Paper no. 90, pages 2580–2616.

Journal URL

http://www.math.washington.edu/~ejpecp/

Recurrence and transience for long range reversible

random walks on a random point process

∗

Pietro Caputo

†

Alessandra Faggionato

‡

Alexandre Gaudillière

§

Abstract

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 recur-rent 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. Key words: random walk in random environment, recurrence, transience, point process, elec-trical network.

AMS 2000 Subject Classification:Primary 60K37; 60G55; 60J45.

Submitted to EJP on January 23, 2009, final version accepted November 3, 2009.

∗_{This work was supported by the European Research Council through the “Advanced Grant” PTRELSS 228032 and by}

the GREFI-MEFI

†_{Dip. Matematica, Università di Roma Tre, L.go S. Murialdo 1, 00146 Roma, Italy. e–mail: caputo@mat.uniroma3.it} ‡_{Dip. Matematica, Università di Roma “La Sapienza”, P.le Aldo Moro 2, 00185 Roma, Italy. e–mail:}

fag-giona@mat.uniroma1.it

1

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 Rd_{, d} >1 and callX_n the discrete time Markov chain with state spaceS that jumps from a site x to another site y with a probability p(x,y)that is proportional to ϕ(_|x − y|), where ϕ:(0,_∞)_→(0, 1]is a positive bounded measurable function and|x|stands for the Euclidean norm ofx ∈Rd_{. We writeP}

for the law ofX_n, so that forx ₆= y_∈S:

P(X_n+1= y_|X_n=x) =p(x,y):= ϕ(|y−x|)

w_S(x) ,

where w_S(x) := P_{z∈S:z=6 x}ϕ(_|z₋ x_|). Note that the random walk X_n is well defined as soon as w_S(x)_∈(0,_∞)for everyx _∈S. In this case,w_S=_{w_S(x), x _∈S_}is a reversible measure:

w_S(x)p(x,y) =w_S(y)p(y,x).

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 functionϕ_p,α(t) :=1∧t−d−α, α > 0 and

the stretched exponential functionϕe,β(t):=exp(−tβ),β >0. We investigate here the transience

and recurrence of the random walkXn. We recall that Xn is said to berecurrentif for some x ∈S, the walk started at X0 = 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 = Zd, 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 [25](see also Appendix B). For instance, it is well known that for dimensiond>3 both(Zd_,ϕe,β)and(Zd_{,ϕp,α)are transient for allβ >0 andα >0}

while ford=1, 2,(Zd_,ϕe,β)is recurrent for allβ >0 and(Zd_,ϕp,α)is transient iff 0< α <d. We shall be interested in the case whereSis a locally finiterandomsubset ofRd, i.e. the realization of a simple point process onRd_{. We denote by}P_{the law of the point process. For this model to be}

well defined forP_{–almost allS we shall require that, given the choice ofϕ:}

P _{wS(x)∈(0,∞), for all x ∈S=1 . (1.1)}

If we look at the setS as a random perturbation of the regular latticeZd_{, the first natural question}

is to find conditions on the law of the point processP_{and the functionϕ such that(S,ϕ)is}P_–a.s.

transient (recurrent) iff(Zd_{,ϕ)is transient (recurrent). In this case we say that the random walks} (S,ϕ) and (Zd_,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 (S,ϕ). Before going to a description of our main results we discuss the main examples of point processes we have in mind. In what follows we shall use the notationS(Λ)

for the number of points ofS in any given bounded Borel setΛ_⊂Rd_{. For anyt>0 andx ∈}Rd _we

write

Q_x,t:=x+

•

−t

2, t 2

˜d

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

R∞

0 t

d−1_{ϕ(t)d t<}

∞. Suppose the law of the point processP_{is such that}

sup x∈Zd

E_{[S(Qx,1)]<∞. (1.2)}

Then it is immediate to check that(S,ϕ)satisfies (1.1) for anyϕ∈Φ_d.

1.1

Examples

The main example we have in mind is the case when P is ahomogeneous Poisson point process

(PPP) onRd_{. In this case we shall show that}(S,ϕ)and(Zd_,ϕ)have a.s. the same type, at least for the standard choicesϕ=ϕp,α,ϕe,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,ϕe,β)model

withβ =1 is a variant of the well known Mott variable–range hopping model, see[12]for more details. The original variable–range hopping model comes with an environment of energy marks on top of the Poisson point process that 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 that 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 setΓ_⊂Rd _such

that, for a suitable basisv₁,v₂, . . . ,v_d ofRd_{, one has}

Γ₋x = Γ _∀x _∈G:=z₁v₁+z₂v₂+_{· · ·}+z_dv_d : z_{i ∈}Z _{∀i . (1.3)}

Thespatially randomized andp–diluted crystalis obtained fromΓby first translatingΓby a random vectorV chosen with uniform distribution in the elementary cell

∆ =t₁v₁+t₂v₂+_{· · ·}+t_dv_d : 06t_i <1 _∀i ,

and then erasing each point with probability 1−p, independently from the others. One can check that the above construction depends only onΓand not on the particular G and∆ chosen. In the case of spatially randomized andp–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 [12] for a discussion on the Palm distribution and its relation to Mott variable–range hopping). Therefore, we shall not explic-itly mention in the sequel, to avoid lengthy repetitions, the application of our estimates to these cases. We shall also comment on applications of our results to two other classes of point processes: percolation clustersanddeterminantal point processes. We say thatS is a percolation cluster whenP

is the law of the infinite cluster in super–critical Bernoulli site–percolation onZd. 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, e.g., [15; 5]. In particular, it is well known that the simple random walk on the percolation

cluster has almost surely the same type as the simple random walk onZd_{. Our results will allow to}

prove that ifS is the percolation cluster onZd then(S,ϕ)has a.s. the same type as(Zd,ϕ), at least for the standard choicesϕ=ϕp,α,ϕe,β. Determinantal point processes (DPP) on the other hand are

defined as follows, see[24; 4]for recent insightful reviews on DPP. Let_K be a locally trace class self–adjoint operator on L2(Rd,d x). If, in addition,_K satisfies 06_K 61 we can speak of the DPP associated withK. LetP_,E_{denote the associated law and expectation. It is always possible to}

associate with_K a kernelK(x,y)such that, for any bounded measurable setB_⊂Rd_{, one has} E[S(B)] =tr(_K1B) =

Z

B

K(x,x)d x<∞ (1.4)

where S(B)is the number of points in the set B and 1_B stands for multiplication by the indicator function of the setB, see[24]. Moreover, for any family of mutually disjoint subsetsD₁,D₂, . . . ,D_k⊂

Rd _{one has}

E

 

k

Y

i=1 S(D_i)



=

Z

Q

iDi

ρ_k(x1,x2, . . . ,xk)d x1d x2. . .d xk, (1.5) where thek–correlation functionρksatisfies

ρ_k(x1,x2, . . . ,xk) =det

€

K(x_i,x_j)Š

16i,j6k.

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 ind =1, whereK(x,y) = sin(π(x−y))

π(x−y) . Our

results will imply for instance that for stationary DPP(S,ϕ)and(Zd,ϕ)have a.s. the same type if

ϕ=ϕp,α (anyα >0) orϕ=ϕe,β withβ <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[9; 20]. 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 setS and complete edge set {{x,y}, x 6= 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 magnitudeϕ(_|x−y|). We point out that other long–range reversible random walks have already been studied (see for example [2], [3], [17], [22]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 ofXnin terms of the associated resistor network. Let {S_n}n>1 be an increasing sequence of subsets Sn ⊂ S such that S = ∪n>1Sn and let (S,ϕ)n denote the network obtained by collapsing all sites inS_nc=S_\S_ninto a single sitez_n(this corresponds to the network where all resistors between nodes inS_nc are replaced by infinitely conducting wires but all other wires connectingSnwithSnandSn withSnc are left unchanged). Forx∈Sandnlarge

enough such that x _∈S_n, letR_n(x) denote the effective resistance between the nodes x andz_n in the network (S,ϕ)_n. We recall that Rn(x) equals the inverse of the effective conductance Cn(x), defined as the current flowing in the network when a unit voltage is applied across the nodes x andz_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 setSn for our original random walk(S,ϕ) started at x (see, e.g., (4.28) in

[13]). The sequenceR_n(x)is non–decreasing and its limitR(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 the light of this, we shall investigate the rate of divergence ofR_n(x) for specific recurrent models. Lower bounds onR_n(x) can be obtained by the following variational characterization of the effective conductanceCn(x):

C_n(x) = inf h:S→[0,1]

h(x)=0 ,h≡1 onS_nc 1 2

X

y,z∈S y6=z

ϕ(_|y₋z_|) h(y)₋h(z)2. (1.6)

The infimum above is attained when hequals the electrical potential, set to be zero on x and 1 on S_nc. From (1.6) one derives Rayleigh’s monotonicity principle: the effective conductance C_n(x)

decreases whenever ϕ is replaced by ϕ′ satisfying ϕ′(t) 6 ϕ(t) for all t > 0. Upper bounds on Rn(x)can be obtained by means of fluxes. We recall that, given a point x ∈S and a subsetB ⊂S not containing x, a unit flux fromx toB is any antisymmetric function f :S×S→R_{such that}

divf(y):=X

z∈S f(y,z)

  

=1 if y =x,

=0 if y ₆=x and y_6∈B,

60 if y _∈B.

If B= _;then f is said to be a unit flux from x to∞. The energyE(f)dissipated by the flux f is defined as

E(f) =1

2

X

y,z∈S y6=z

r(y,z)f(y,z)2. (1.7)

To emphasize the dependence onS andϕwe shall often call_E(f)the(S,ϕ)–energy. Finally,R_n(x), R(x)can be shown to satisfy the following variational principles:

R_n(x) =inf¦_E(f) : f unit flux from x toS_nc©, (1.8) R(x) =infE(f) : f unit flux fromx to∞ . (1.9) In particular, one has the so called Royden–Lyons criterion[19]for reversible random walks: the random walkX_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, which we shall often use in the sequel.

Lemma 1.1. LetP_,P′_{denote two point processes on}Rd _{such that}P_{is stochastically dominated by}P′

and letϕ,ϕ′:(0,∞)_→(0,∞)be such thatϕ 6Cϕ′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 toP_andP′_{, respectively. The following holds:}

1. if(S,ϕ)is transientP_{–a.s., then(S}′_,ϕ′_{)is transient}P′_–a.s.

2. if(S′,ϕ′)is recurrentP′_{–a.s., then}(S,ϕ)is recurrentP_–a.s.

Proof. The stochastic domination assumption is equivalent to the existence of a coupling ofP_andP′

such that, almost surely,S_⊂S′(see e.g.[14]for more details). If(S,ϕ)is transient then there exists a flux f onS with finite(S,ϕ)–energy from some x _∈S to infinity. We can lift f to a flux onS′_⊃S (from the same x to infinity) by setting it equal to 0 across pairs x,y where eitherx or y (or both) are not inS. This has finite(S′,ϕ)-energy, and sinceϕ6Cϕ′it will have finite(S′,ϕ′)–energy. This proves (1). The same argument proves (2) since if S _⊂ S′ were such that(S,ϕ) is transient then

(S′,ϕ′)would be transient and we would have a contradiction.

1.3

General results

Recall the notationB_x,t for the open ball inRd _{centered at x with radius t and define the function}

ψ:(0,_∞)_→[0, 1]by

ψ(t):= sup x∈Zd

P _{S Bx,t}=0. (1.10)

Theorem 1.2. (i) Let d>3andα >0, or d=1, 2and0< α <d. Suppose thatϕ∈Φ_d and

ϕ(t)>cϕp,α(t), (1.11)

ψ(t)6C t−γ, ∀t>0 , (1.12)

for some positive constants c,C andγ >3d+α. Then, P_{–a.s. (S,ϕ) is transient. (ii) Suppose that}

d>3and _Z

∞

0

ea tβψ(t)d t<∞, (1.13)

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−δtβ for some c>0. (iii) Set d>1and suppose that sup

x∈Zd

E”_S(Q_x,1)2—<∞. (1.14)

Then(S,ϕ)isP_{–a.s. recurrent whenever(}Zd_{,ϕ0)is recurrent, whereϕ0 is given by}

ϕ0(x,y):= max u∈Qx,1,v∈Qy,1

ϕ(_|v−u|). (1.15)

The proofs of these general statements are given in Section 2. It relies on rather elementary ar-guments not far from therough embedding method described in[20, 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 onZd_{. Point (iii) of Theorem 1.2 is obtained by exhibiting}

a candidate for the electric potential in the network(S,ϕ)that produces a vanishing conductance. Again the construction is achieved using comparison with long–range random walks onZd_.

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 withϕ=ϕp,α,α <2. More

Corollary 1.3. Fix d>1. LetP_{be one of the following point processes: a homogeneous PPP; the infinite}

cluster in super–critical Bernoulli site–percolation onZd_{; a stationary DPP on} Rd_{. Then}(S,ϕp,α)has

a.s. the same type as(Zd_,ϕp,α), for allα >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 being 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 whenϕ 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 forme−c td−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)>C e−a td−1. However, the well known Grimmett–Kesten–Zhang theorem about the transience ofnearest–neighborrandom walk on the infinite cluster ind>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–percolationifP_{is such that for some L∈}N_{the random field}

σ= σ(x) : x _∈Zd_{, σ(x):=χ S(QL x,L)>1, (1.16)}

stochastically dominates the i.i.d. Bernoulli field onZd _{with some super–critical parameterp. Here}

χ(_·)stands for the indicator function of an event. In particular, it is easily checked that any homo-geneous PPP dominates super–critical Bernoulli site–percolation. For DPP defined onZd_{, stochastic}

domination w.r.t. Bernoulli can be obtained under suitable hypotheses on the kernel K, see[21]. We are not aware of analogous conditions in the continuum that would imply that DPP dominates super–critical Bernoulli site–percolation. 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 forϕ=ϕe,β in the following

Corollary 1.4. 1. Let P _{be any of the processes considered in Corollary 1.3. Then, for any β > 0,} (S,ϕe,β) is a.s. recurrent when d = 1, 2. 2. Let P be the law of the infinite cluster in super–critical

Bernoulli site–percolation on Zd _{or a homogeneous PPP or any other process that dominates super–}

critical Bernoulli site–percolation. Then, for anyβ >0,(S,ϕe,β)is a.s. transient when d>3. 3. LetP

be any stationary DPP. Then, for anyβ∈(0,d),(S,ϕe,β)isP–a.s. transient when d>3

We point out that, by the same proof, statement 2) above remains true if(S,ϕe,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 with the finite sets S_n:=S_∩[₋n,n]ddiverge, 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, withϕ = ϕp,α, α ∈[1,∞), and the case d =2 with ϕ = ϕp,α, α ∈[2,∞). By Rayleigh’s

particular, they cover the stretched exponential case (S,ϕe,β). We say that the point process P is

dominated by an i.i.d. fieldif the following condition holds: There existsL∈Nsuch that the random field

N_L= N(v) : v_∈Zd_{, N(v):=S(QL v,L),}

is stochastically dominated by independent non–negative random variables_{Γ_v, v_∈Zd_{}with finite}

expectation. For the results in dimensiond=1 we shall require the following exponential moment condition on the dominating fieldΓ: There existsǫ >0 such that

E[eǫΓv_{]<∞. (1.17)}

For the results in dimensiond=2 it will be sufficient to require the existence of the fourth moment:

E”_Γ4

v

—

<∞. (1.18)

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 fieldΓ satisfies (1.17). Moreover, this continues to hold for non–homogeneous Poisson process with a uniformly bounded intensity function. We refer the reader

to[21; 14]for examples of determinantal processes satisfying this domination property.

Theorem 1.5. Set d =1, ϕ=ϕp,α andα>1. Suppose that the point processPis dominated by an

i.i.d. field satisfying (1.17). Then, forP_{–a.a. S the network} (S,ϕ)satisfies: given x∈S there exists a constant c>0such that

Rn(x)>c

      

logn ifα=1 , nα−1 if1< α <2 , n/logn ifα=2, , n ifα >2 ,

(1.19)

for all n>2such that x∈Sn.

Theorem 1.6. Set d=2,ϕ=ϕp,α andα>2. Suppose thatPis dominated by an i.i.d. field satisfying

(1.18). Then, forP_{–a.a. S the network (S,ϕ) satisfies: given x ∈S there exists a constant c>0such}

that

R_n(x)>c

(

logn ifα >2 ,

log(logn) ifα=2 , (1.20)

for all n>2such that x_∈S_n.

The proofs of Theorem 1.5 and Theorem 1.6 are 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 re-duction of the network that ultimately leads to a one–dimensional nearest–neighbor network where effective resistances are easier to estimate. This construction uses an idea that already appeared in

[18]. – see also[7]and[1]for recent applications. The construction allows to go from long–range

to nearest–neighbor networks, as explained in Section 3. 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(Zd_,ϕp,α)network. In particular, the method

can be used to obtain quantitative lower bounds onR_n(x)for the deterministic caseS_≡Zd_{, which}

is indeed a special case of the theorems. In the latter case the lower bounds obtained here, as well as suitable upper bounds, are probably well known but we were not able to find references to them in the literature. In appendix B, we show how to bound from above the effective resistanceR_n(x)of the network(Zd,ϕp,α)by means of harmonic analysis. The resulting upper bounds match 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 factorplogn.

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(Zd,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 criterion 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_,ϕp,α) is transient forα <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(Z2,ϕp,α)is transient for

α <2. In order to state the one-dimensional result, it is convenient to number the points ofS as S=_{x_i}i∈I where x_i < x_i+1, x₋₁ <06x₀ andN_{⊂ I or−}N_{⊂I (we assume that|S|=∞,}P_–a.s.,

since otherwise the network is recurrent). For simplicity of notation we assume below thatN_⊂I,

P_{–a.s. The following result can be easily extended to the general case by considering separately}

the conditional probabilitiesP_(·|N_{⊂ I)and}P_(·|N_{6⊂ I), and applying a symmetry argument in the}

second case.

Proposition 1.7. Take d = 1 and α ∈ (0, 1). Suppose that the following holds for some positive constants c,C:

ϕ(t)>cϕp,α(t), t>0 , (1.21)

E _|xn−xk|1+α6C(n₋k)1+α, _∀n>k>0 . (1.22) ThenP_{–a.s.(S,ϕ)is transient. In particular, if}P_{is a renewal point process such that}

E_(|x1−x0|1+α_{)<∞, (1.23)}

thenP_–a.s.(S,ϕ)is transient.

Suppose thatP_{is a renewal point process and write}

e

ψ(t):=P(x1−x0>t).

Then (1.23) certainly holds as soon as e.g.ψesatisfiesψe(t)6C t−(1+α+ǫ)for some positive constants C,ǫ. We can check that this improves substantially over the requirement in Theorem 1.2, part (i), since ifψis defined by (1.10), then we have, for allt>1:

˜

ψ(2t) =P(S∩Bx0+t,t=;)6ψ(t−1).

IdentifyR2 _{with the complex plane}C_{, and define the setS}

∗:=∪∞n=0Cn, where C_n:=nnekn2+iπ1 ∈C: k∈ {0, . . . ,n}

o

. (1.24)

Theorem 1.8. The network(S_∗,ϕp,α)is transient for allα∈(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(Z2,ϕp,α)forα∈(0, 2). The proofs

of Proposition 1.7 and Theorem 1.8 are given in Section 4.1.

2

Recurrence and transience by comparison methods

LetS0 be a given locally finite subset of Rd and let(S0,ϕ0)be a random walk on S0. We assume thatw_S0(x)<∞for all x ∈S0 and thatϕ0(t)>0 for allt >0. Recall that in the resistor network picture every node {x,y} is given the resistance r0(x,y) := ϕ0(|x − y|)−1. To fix ideas we may think ofS₀=Zd _{and eitherϕ0 =ϕp,α orϕ0=ϕe,β. (S0,ϕ0)will play the role of the deterministic}

background network.

LetP_{denote a simple point process on}Rd_{, i.e. a probability measure on the set}Ω of locally finite subsetsS ofRd_{, endowed with theσ–algebraF generated by the counting mapsNΛ:Ω→}N_{∪ {0},}

whereN_Λ(S) =S(Λ)is the number of points ofS that belong toΛandΛis a bounded Borel subset ofRd_{. We shall useSto denote a generic random configuration of points distributed according to}P_.

We assume thatP_{andϕ are such that (1.1) holds. Next, we introduce a mapφ:S0→S, from our}

reference setS₀to the random setS. For anyx _∈S₀we writeφ(x) =φ(S,x)for the closest point in Saccording to Euclidean distance. If the Euclidean distance from x toS is minimized by more than one point inS then chooseφ(x)to be the lowest of these points according to lexicographic order. This defines a measurable mapΩ_∋S_7→φ(S,x)_∈Rd _{for every x ∈S0. For any pointu∈S define}

thecell

V_u:=_{x _∈S₀ : u=φ(x)_}.

By construction_{V_u, u_∈S_} determines a partition of the original vertex set S₀. Clearly, some of theVu may be empty, while some may be large (ifS has large “holes” with respect toS0). Let N(u) denote the number of points (ofS₀) in the cellV_u. We denote byE_{the expectation with respect to} P_.

Lemma 2.1. Suppose(S0,ϕ0)is transient. If there exists C <∞such that, for all x6= y in S0,

E_{N(φ(x))N(φ(y))r(φ(x),φ(y))} 6C r₀(x,y), (2.1) then(S,ϕ)isP_{–a.s. transient.}

Proof. Without loss of generality we shall assume that 0 _∈ S₀. Since (S₀,ϕ0) is transient, from the Royden–Lyons criterion recalled in Subsection 1.2, we know that there exists a unit flux f : S0×S0→Rfrom 0∈S0 to∞with finite(S0,ϕ0)-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 anyu,v_∈S we define

θ(u,v) = X

x∈Vu

X

y∈Vv

If eitherV_u or V_v are empty we setθ(u,v) =0. Note that the above sum is finite for all u,v _∈S,

P–a.s. Indeed condition (2.1) implies thatN(φ(x))<∞for all x∈S0,P–a.s. Thus,θdefines a unit flux fromφ(0)to infinity on(S,ϕ). Indeed, for everyu,v ∈S we haveθ(u,v) =₋θ(v,u)and for everyu₆=φ(0)we haveP_v∈Sθ(u,v) =0. Moreover,

X

v∈S

θ(φ(0),v) = X

x∈Vφ(0) X

y∈S0

f(x,y) = X

x∈Vφ(0):

x6=0

X

y∈S0

f(x,y) + X

y∈S0

f(0,y) =0+1=1 .

The energy of the fluxθ is given by

E(θ):= 1

2

X

u∈S

X

v∈S

θ(u,v)2r(u,v). (2.2)

From Schwarz’ inequality

θ(u,v)26N(u)N(v)X

x∈Vu

X

y∈Vv

f(x,y)2. It follows that

E(θ)61

2

X

x∈S0

X

y∈S0

f(x,y)2N(φ(x))N(φ(y))r(φ(x),φ(y)). (2.3) Sincef has finite energy on(S0,ϕ0)we see that condition (2.1) impliesE[E(θ)]<∞. In particular, this shows that P_{–a.s. there exists a unit fluxθ from some point u0 ∈S to∞ with finite (S,ϕ)}

-energy. To produce an analogue of Lemma 2.1 in the recurrent case we introduce the seteS=S∪S0 and consider the network(eS,ϕ). From monotonicity of resistor networks, recurrence of(S,ϕ)is implied by recurrence of(eS,ϕ). We define the mapφ′:eS→S0, fromSeto the reference setS₀ as the mapφ introduced before, only with S₀ replaced bySeandS replaced byS₀. Namely, given x ∈Sewe defineφ′(x)as the closest point inS0 according to Euclidean distance (when there is more than one minimizing point, we take the lowest of these points according to lexicographic order). Similarly, for any pointx _∈S₀we define

V_x′:=_{u_∈eS : x=φ′(u)_}.

Thus _{V_x′, x ∈S0} determines a partition of Se. Note that in this case all Vx′ are non–empty (Vx′ containsx ∈Se). As an example, ifS0=Zd andx ∈eS, thenφ′(x)is the only point y inZd such that x _∈y+ (₋1/2, 1/2]d, whileV_x′=eS_∩(x+ (₋1₂,1₂]d)for anyx _∈Zd_.

Lemma 2.2. Suppose that(S₀,ϕ0)is recurrent and thatP–a.s. Vx′is finite for all x∈S0. If there exists C <∞such that, for all x6= y in S0,

Eh X

u∈V′

x

X

v∈V′

y

ϕ(_|u₋v_|)i6Cϕ0(|x− y|), (2.4)

then(S,ϕ)isP_{–a.s. recurrent.}

Proof. Without loss of generality we shall assume that 0_∈S₀. SetS_0,n=S_0∩[₋n,n]d, collapse all sites inS_0,nc =S0\S0,n into a single sitezn and callc(S0,n)the effective conductance between 0 and

z_n, i.e. the net current flowing in the network when a unit voltage is applied across 0 andz_n. Since

(S0,ϕ0)is recurrent we know thatc(S0,n)→0,n→ ∞. Recall thatc(S0,n)satisfies c(S_0,n) = 1

2

X

x,y∈S0

ϕ0(|x−y|)(ψn(x)−ψn(y))2, (2.5) whereψnis the electric potential, i.e. the unique function on S0 that is harmonic inS0,n, takes the value 1 at 0 and vanishes outside ofS0,n. GivenS∈Ω, set

e

S_n=_∪x∈S0,nV_x′.

Note thatSen is an increasing sequence of finite sets, covering all S. Collapse all sites in(eSn)c into a single siteez_n and callc(Se_n)the effective conductance between 0 andez_n (by construction 0∈Se_n). From the Dirichlet principle (1.6) we have

c(Sen)6 1 2

X

u,v∈eS

ϕ(_|u−v|)(g(u)₋g(v))2,

for any g:eS→[0, 1]such thatg(0) =1 andg=0 on(Se_n)c_{. Choosingg(u) =ψ}

n(φ′(u))we obtain c(Sen)6

1 2

X

x,y∈S0

(ψn(x)−ψn(y))2

X

u∈V′

x

X

v∈V′

y

ϕ(_|u−v|).

From the assumption (2.4) and the recurrence of (S₀,ϕ0) implying that (2.5) goes to zero, we deduce thatE[c(Sen)]→0,n→ ∞. Since c(Sen)is monotone decreasing we deduce thatc(Sen)→0,

P_{–a.s. This implies the}P_{–a.s. recurrence of}(Se,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 S₀ = Zd _{andϕ0 = ϕp,α. Since (S0,ϕp,α) is transient whenever d} > 3, or d = 1, 2 and 0< α <d (the classical proof of these facts through harmonic analysis is recalled in Appendix B), we only need to verify condition (2.1). For the moment we only suppose that ψ(t) 6 C′t−γ for someγ > 0. Let us fix p,q >1 s.t. 1/p+1/q =1. Using Hölder’s inequality and then Schwarz’ inequality (or simply using Hölder inequality with the triple(1/2q, 1/2q, 1/p)), we obtain

E_N(φ(x))N(φ(y))r(φ(x),φ(y)) (2.6)

6E”_N(φ(x))2q— 1

2q_E”_N(φ(y))2q— 1

2q_{Er(φ(x),φ(y))}p 1 p

for anyx ₆= y inZd_{. By assumption (1.11) we know that}

r(φ(x),φ(y))p6c rp,α(φ(x),φ(y))p:=c 1∨ |φ(x)−φ(y)|p(d+α)

. (2.7)

We shall usec₁,c₂, . . . to denote constants independent ofx and y below. From Jensen’s inequality

|φ(x)₋φ(y)_|p(d+α)6c1

€

From (2.7) and the fact that_|x₋y_|>1 we derive that

E_{r(φ(x),φ(y))}p 6c₂sup z∈Zd

E”_|φ(z)−z|p(d+α)—_+c2|x−y|p(d+α)_{. (2.8)}

Now we observe that_|φ(z)₋z_|>t if and only ifB_z,t∩S=_;. Hence we can estimate

E”_|φ(z)−z|p(d+α)— 61+

Z ∞

1

ψ



t

1 p(d+α)

‹

d t61+C

Z ∞

1 t−

γ

p(d+α)_{d t6c}

3,

provided thatγ >p(d+α). Therefore, using|x− y|>1, from (2.8) we see that for any x 6= y in

Zd_:

E_{r(φ(x),φ(y))}p 1 p _6c

4rp,α(x,y). (2.9)

Next, we estimate the expectationE”_N(φ(x))2q—_{from above, uniformly in x∈}Zd_{. 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) ₆= _; and S∩B(x ±3pd t ei,t) 6= ;, where

{e_i : 16i6d}is the canonical basis ofRd_{. Then, on the event E}(x,t)we haveφ(x)_∈B(x,t), i.e.

|φ(x)₋x_|<t, and z_6∈V_φ(x)for all z_∈Rd _{such that|z−x|>9d}p_{d t}

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 c₅, the event N(φ(x)) > c₅td implies that at least one of the 2d+1 ballsB(x,t),B(x±3pd t ei,t)must have empty intersection withS. SinceB(x±⌊3

p

d t⌋ei,t− 1)_⊂B(x_±3pd t e_i,t)fort>1, we conclude that

P”_N(φ(x))>c5td

—

6(2d+1)ψ(t−1), t>1. Takingc6 such thatc

−1_d

5 c

1 2qd

6 =2, it follows that

E”_N(φ(x))2q—=

Z ∞

0

P(N(φ(x))2q>t)d t

6c₆+ (2d+1)

Z ∞

c6

ψ

c−

1 d

5 t

1 2qd ₋₁

d t6c₆+c₇

Z ∞

1 t−

γ

2qd_{d t6c}

8, (2.10)

as soon asγ >2qd. Due to (2.6), (2.9) and (2.10), the hypothesis (2.1) of Lemma 2.1 is satisfied whenψ(t)6C t−γfor all t>1, whereγis a constant satisfying

γ >p(d+α), γ >2qd= 2pd

p₋1.

We observe that the functions (1,_∞) _∋ p _7→ p(d +α) and (1,_∞) _∋ p _7→ 2pd

p−1 are respectively increasing and decreasing and intersect in only one pointp_∗=3d+α

d+α. Hence, optimizing overp, it is

enough to require that

γ > inf

p>1max{p(d+α), 2pd

This concludes the proof of Theorem 1.2 (i). Proof of lemma 2.3. The first claim is trivial since S_∩B(x,t)₆= _; implies φ(x) _∈B(x,t). In order to prove the second one we proceed as follows. For simplicity of notation we setm:=3pd andk:=9d. Let us takez_∈Rd _{with|z−x|>k}p_{d t.}

Without loss of generality, we can suppose that x =0, z1 >0 and z1 >|zi| for all i =2, 3, . . . ,d. Note that this implies thatkpd t<|z_|6pdz₁, hencez₁>kt. Since

min

u∈B(0,t)|z−u|=|z| −t

max u∈B(mt e1,t)

|z−u|6_|z−mt e1|+t, if we prove that

|z_{| − |}z₋mt e_1|>2t, (2.11) we are sure that the distance fromz to each point inS∩B(0,t)is larger than the distance fromz to each point ofS_∩B(mt e₁,t). Hence it cannot be thatz_∈V_φ(0). In order to prove (2.11), we first observe that the map(0,_∞)_∋ x 7→px+a−px+b∈(0,_∞)is decreasing fora> b. Hence we obtain that

|z_{| − |}z₋mt e_1|>

p

dz₁₋Æ(z₁₋mt)2_{+ (d−1)z}2 1. Therefore, settingx :=z₁/t, we only need to prove that

p

d x−p(x−m)2_{+ (d−1)x}2_{>2 , ∀x >k.} By the mean value theorem applied to the function f(x) =px

p

d x−p(x−m)2_{+ (d−1)x}2_> 1 2pd x

€

d x2−(x−m)2₋(d−1)x2Š=

2x m−m2 2pd x >

m

p

d − m2

k =2 . This completes the proof of (2.11).

Proof of Theorem 1.2 Part (ii). We use the same approach as in Part (i) above. We start again our estimate from (2.6). Moreover, as in the proof of (2.10) it is clear that hypothesis (1.13) implies

E[N(φ(x))2q_{]<∞for anyq>1, uniformly in x∈Z}d_{. Therefore it remains to check that} r₀(x,y):=E_{r(φ(x),φ(y))}p

1

p _{, (2.12)}

defines a transient resistor network onZd_{, for anyd}>3, under the assumption that r(φ(x),φ(y))6C eδ|φ(x)−φ(y)|β.

For anyβ >0 we can find a constantc₁=c₁(β)such that

r(φ(x),φ(y))6C exp€δc₁”_|φ(x)₋x_|β+_|x₋y_|β+_|φ(y)₋ y_|β—Š. Therefore, using Schwarz’ inequality we have

E_r(φ(x),φ(y))p1p

6c₂exp€δc_2|x₋y_|βŠE”_exp€_{δc2|φ(x)−x|}βŠ— 1

2_E”_exp€_δc

2|φ(y)−y|β

Š—1 2 _.

Forγ >0

E”_exp€_γ|φ(x)₋x|βŠ—61+

Z ∞

1

ψ 1

γ logt

1

β

!

d t

=1+γβ

Z ∞

0

ψ(s)eγsβsβ−1ds,

where, using (1.13), the last integral is finite forγ <a. Takingγ=δc₂ andδsufficiently smal we arrive at the conclusion that uniformly inx,

E_{r(φ(x),φ(y))}p 1 p _6c

3exp

€

c_3|x₋y_|βŠ=:_er₀(x,y).

Clearly,_er₀(x,y)defines a transient resistor network onZd _{and the same claim for r0(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 S₀=Zd. With this choice ofS0 we have thatVx′⊂ {x} ∪(S∩Qx,1), x∈Zd. Recalling definition (1.15) we see that for allx ₆= y inZd_,

E

  

X

u∈V′

x

X

v∈V′

y

ϕ(_|u−v|)

 

6c₁ϕ₀(x,y)E”(1+S(Q_x,1))(1+S(Q_y,1))—.

Using Schwarz’ inequality and condition (1.14) the last expression is bounded above byc2ϕ0(x,y). This implies condition (2.4) and therefore the a.s. recurrence of(S,ϕ).

2.2

Proof of Corollary 1.3

We start with some estimates on the functionψ(t). Observe that, for Poisson point processes PPP(λ), one hasψ(t) = e−λtd. A similar estimate holds for a stationary DPP. More generally, for DPP we shall use the following facts.

Lemma 2.4. Let P _{be a determinantal point process on} Rd _{with kernel K. Then the function ψ}(t)

defined in (1.10) equals

ψ(t) = sup x∈Zd

Y

i

1₋λi(B(x,t))

, (2.13)

whereλi(B)denote the eigenvalues ofK1Bfor any bounded Borel set B⊂Rd. In particular, condition (1.12) is satisfied if

exp

n

−

Z

B(x,t)

K(u,u)du

o

6C t−γ, t>0 , x ∈Zd_{, (2.14)}

for some constants C>0andγ >3d+α. IfP_{is a stationary DPP thenψ(t)6e}−δtd_{, t>0, for some}

δ >0. Finally, condition (1.14) reads sup

x∈Zd

nX

i

λi(Q(x, 1)) +

X

i

X

j:j6=i

λi(Q(x, 1))λj(Q(x, 1))

o

<∞. (2.15)

Proof. It is known (see [4] and [24]) that, for each bounded Borel set B _⊂ Rd_{, the number of}

pointsS(B)has the same law as the sumP_iBi,Bi’s being independent Bernoulli random variables with parameters λ_i(B). This implies identity (2.13). Since 1−x 6 e−x, x > 0, we can bound the r.h.s. of (1.10) byQ_ie−λi(B) _{= e}−T r(K1B)_{. This identity and (1.4) imply (2.14). If the DPP is}

stationary thenK(u,u)_≡K(0)>0 and thereforeψ(t)6e−K(0)td. Finally, (2.15) follows from the identityE[S(Q(x, 1))2] =P_iλi(Q(x, 1)) +

P

i

P

j:j6=iλi(Q(x, 1))λj(Q(x, 1)), again a consequence of the fact that S(Q(x, 1)) is the sum of independent Bernoulli random variables with parameters

λi(Q(x, 1)). Since the sum of theλi(Q(x, 1))’s is finite,E[S(Q(x, 1))2]<∞for any x ∈Zd. If the DPP is stationary it is uniformly finite. The next lemma allows to estimateψ(t)in the case of percolation clusters.

Lemma 2.5. LetP_{be the law of the infinite cluster in super–critical Bernoulli site (or bond) percolation}

inZd_{, d>2. Then there exist constants k,δ >0such that}

e−δ−1nd−16ψ(n)6k e−δnd−1, n_∈N_.

Proof. The lower bound follows easily by considering the event that e.g. the cube centered at the origin with sidenhas all the boundary sites (or bonds) unoccupied. To prove the upper bound one can proceed as follows. LetK_n(γ),γ >0, denote the event that there exists an open clusterC inside the box B(n) = [₋n,n]d_∩Zd such that_|C_|>γnd. 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[23]ford >3) imply that there exist constantsk₁,δ1,γ >0 such that

P_(Kn(γ)c₎6k₁e−δ1nd−1_{. (2.16)}

On the other hand, let C_x denote the open cluster at x _∈Zd _{and writeC∞ for the infinite open}

cluster. From [16, Theorem (8.65)] we have that there exist constants k₂,δ2 such that for any x ∈Zd _{and for anyγ >0:}

P_(γnd6_|C_x|<∞)6k₂e−δ2nd−1_{. (2.17)}

Now we can combine (2.16) and (2.17) to prove the desired estimate. For anynwe write

P_{(B(n)∩ C∞=;)}6P_{(B(n)∩ C∞=;; Kn(γ)) +}P_(Kn(γ)c_).

The last term in this expression is bounded using (2.16). The first term is bounded by

P_{(∃x ∈B(n) : γn}d _6|Cx|<∞)6 X

x∈B(n)

P_(γnd _6|Cx|<∞).

Using (2.17) we arrive atP(B(n)_{∩ C∞}=_;)6k e−δnd−1for suitable constantsk,δ >0. 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(Zd_,ϕ0)and (Zd,ϕp,α)have the same type, whenϕ0is defined by (1.15) with ϕ=ϕp,α. This ends the proof of

B.1

Recurrence and transience

It is known that X is transient ifd >3 for anyα >0 and in d = 1, 2 it is transient if and only if 0< α < d. Let us briefly recall how this can be derived by simple harmonic analysis. From[25][ Section 8, T1]the random walk is transient ifd >3 for anyα >0, and it is recurrent in d=1 for

α >1 and in d = 2 forα > 2. Other cases are not covered by this theorem but one can use the following facts. Define the characteristic function

φ(θ) = X

x∈Zd

p(x)ei x·θ.

Note thatφ(θ)is real, ₋16φ(θ)61, andφ(θ)<1 for allθ 6=0. By the integrability criterion given in[25][Section 8, P1],X is transient if and only if

lim

t↑1

Z

[−π,π)d 1

1₋tφ(θ)dθ <∞. (B.2)

Ifα∈(0, 2)we have, for anyd>1: lim |θ|→0

1₋φ(θ)

|θ|α =κd,α∈(0,∞). (B.3)

The limit (B.3) is proved in[25][Section 8, E2]in the cased=1 but it can be generalized to any d>1. Indeed, writingθ=ǫθˆ,_|θˆ|=1:

1−φ(θ)

|θ|α =ǫ d X

x∈Zd

(1+_|x_|)d+αp(x)1−cos(ǫx· ˆ

θ)

(ǫ+_|ǫx|)d+α , (B.4)

and whenǫ→0, using(1+_|x_|)d+αp(x) =c, we have convergence to the integral c

Z

Rd

f(x)d x, f(x):= 1−cos(x· ˆ

θ)

|x_|d+α ,

whereθˆis a unit vector (the integral does not depend on the choice ofθˆ). This integral is positive and finite forα∈(0, 2) and (B.3) follows. Using (B.3) the integrability criterion (B.2) implies that ford =1 the RW is transient if and only ifα∈(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 [25]. However, one can modify the argument above to obtain that for anyd>1,α=2:

lim |θ|→0

1−φ(θ)

|θ|2_log(|θ|−1₎=κd,2∈(0,∞). (B.5) Thus, using again the integrability criterion (B.2), we see that the case with α= 2 and d = 2 is recurrent. To prove (B.5) one can write, reasoning as in (B.4): For anyδ >0

1₋φ(θ)

|θ|2_log(|θ|−1₎= cǫd

log(ǫ−1₎

X x∈Zd_{: 16|x|6ǫ}−1_δ

1₋cos(ǫx_·θˆ) (ǫ+_|ǫx_|)d+2 +O

€

1/log(ǫ−1)Š

= c

2 log(ǫ−1₎

Z

ǫ6_|x|6δ

x₁2d x

|x_|d+2 +O

€

1/log(ǫ−1)Š,

where x₁ is the first coordinate of the vector x = (x₁, . . . ,x_d). The integral appearing in the first term above is, apart from a constant factor, Rδ

ǫ r

−1_{d r = log(ǫ}−1_{) +const. This proves the claim} (B.5).

B.2

Effective resistance estimates

LetRn :=Rn(0)be the effective resistance associated to the box{x ∈Zd, kxk∞6n}. As already discussed in the introduction, 1_cR_n(wherec>0 is the constant in (B.1)) equals the expected number of visits to the origin before visiting the set_{x_∈Zd_{, kxk}

∞>n}for the random walkX withX0=0. We are going to give upper bounds on Rn in the recurrent cases d = 1, 2, α > min{d, 2}. By

comparison with the simple nearest–neighbor random walk we have that (for anyα)R_n 6Clogn if d =2 andR_n 6C nif d =1. Due to Theorems 1.5 and 1.6, this estimate is of the correct order wheneverp(x)has finite second moment (α >2). The remaining cases are treated as follows. We claim that for some constantC

R_n6C

Z

[−π,π)d

dθ

n−α_{+ (1−φ(θ))}. (B.6)

The proof of (B.6) is given later. Assuming (B.6), we obtain the following bounds:

R_n6C

      

logn d=1,α=1 nα−1 d=1,α∈(1, 2) n/plogn d=1,α=2 log logn d=2,α=2 .

(B.7)

With the only exception of the cased =1,α=2, the upper bounds above and the lower bounds of Theorems 1.5 and 1.6 are of the same order.

The bounds above are easily obtained as follows. For α ∈ [1, 2), d = 1, using the bound 1₋

φ(θ) >λ|θ|α, cf. (B.3), we see that the first two estimates in (B.7) follow by decomposing the integral in (B.6) in the regions|θ|6n−1, |θ|> n−1 and then using obvious estimates. Forα=2, we decompose the integral in (B.6) in the regions _|θ| 6 ǫ, _|θ| > ǫ, ǫ := 1/10. Since 1₋φ(θ) vanishes only forθ = 0, the integral over the region _|θ| > ǫ is of order 1, while we can use the bound 1−φ(θ)>λ|θ|2_log(|θ|−1_{)over the region|θ|6ǫ, cf. (B.5). Hence, we see that for someC}

R_n6C

Z

[−ǫ,ǫ)d

dθ

(n−2_+|θ|2_log(|θ|−1₎₎. (B.8) Then, ifd=1 (B.8) yields

R_n62C

Z (nplogn)−1 0

dθ

n−2 +2C

Z ǫ

(nplogn)−1

dθ θ2_log(θ−1₎. The first integral gives 2C pn

logn. With the change of variables y=1/θthe second integral becomes Z nplogn

ǫ−1

d y logy .

This gives an upper bound O(n/plogn). (Indeed, for ǫ = 1/10 we have, for every y > ǫ−1, (logy)−1 6 2[(logy)−1 ₋ (logy)−2] = 2 d

d y y

R_n 6 C n/plogn in the case d = 1,α = 2. Reasoning as above, if d = 2 and α = 2 we have, for someC:

Rn6C Z ǫ

0

θdθ

(n−2_+θ2_log(θ−1₎₎. We divide the integral as before and obtain

R_n6C n2

Z (nplogn)−1

0

θdθ+C

Z ǫ

(nplogn)−1

dθ θlog(θ−1₎. The first integral is small and can be neglected. The second integral is the same as

Z nplogn ǫ−1

d y

ylogy 6Clog logn. This proves thatRn6Clog logn.

B.3

Proof of claim (B.6)

To prove (B.6) we introduce the truncated kernel Qn(x,y) =Px(X1= y;|X1−x|6c1n) =

c

1+_|y₋x_|d+α1{|y−x|6c1n},

wherec>0 is defined in (B.1) andc1>0 is another constant. Clearly, for all sufficiently largec1 R_n6c

∞

X k=0

Qk_n(0, 0),

whereQk_n(0, 0)is the probability of returning to the origin afterksteps without ever taking a jump of size larger thanc1n. Note that for any x

un:= X y∈Zd

Qn(x,y) =P0(|X1|6c1n) =1−γn, γn:= X

|x|>c1n

p(x)_∼n−α.

LetQˆ_n(x,y)denote the kernel of the RW onZd _{with transition}p(x)conditioned to take only jumps of size less thanc₁n, so thatQˆ_n(x,y) =u−_n1Q_n(x,y). Set

φn(θ) = X x∈Zd

Q_n(0,x)eiθ·x, φˆn(θ) = X x∈Zd

ˆ

Q_n(0,x)eiθ·x.

φ_n(θ) = u_nφˆ(θ) is real and eiθ·x can be replaced by cos(θ·x) in the above definitions. We can write

∞

X k=0

Qk_n(0, 0) = ∞

X k=0

1 (2π)d

Z

[−π,π)d

φn(θ)kdθ=

1 (2π)d

Z

[−π,π)d dθ

1−φn(θ)

, where we use the fact that_|φn(θ)|6un<1 for anyn. Moreover

Therefore it is sufficient to prove that

u_n(1₋φˆn(θ))>δ(1−φ(θ)), (B.9)

for some constantδ >0. LetB(t)denote the Euclidean ball of radius t >0. Supposeθ = y n−1θˆ

for some y>0 and a unit vectorθˆ. Then, un(1−φˆn(θ))

1−φ(θ) =

P

x∈Zd_∩B(c

1n)

1−cos(y(x/n)·θˆ) (n−1_+|(x/n)|)d+α

P

x∈Zd 1−

cos(y(x/n)·θˆ) (n−1_+|(x/n)|)d+α

(B.10)

Reasoning as in the proofs of (B.3) and (B.5) we see that, for allα∈(0, 2], the expression (B.10) is bounded away from 0 for y∈(0,C], fornlarge enough. Indeed, ifα∈(0, 2)we have convergence,

asn_{→ ∞}to _R

B(c1)

1−cos(y x·θˆ) |x|d+α d x

R

Rd

1−cos(y x·θˆ) |x|d+α d x

.

On the other hand, forα=2, from the proof of (B.5) we see that (B.10) converges to 1. Therefore, in all cases (B.9) holds for any_|θ|6C n−1, for all nsufficiently large. Next, we consider the case

|θ|>C n−1. For this range ofθ we know that

1₋φ(θ)>λ|θ|α>λCαn−α,

for someλ >0. Note that this holds also in the caseα=2 according to (B.5). From

φ(θ)₋u_nφˆn(θ) = X x:|x|>c1n

p(x)cos(θ·x),

we obtainφ(θ)>u_nφˆn(θ)−γn. Therefore, for|θ|>C n−1

u_n(1₋φˆn(θ))−δ(1−φ(θ))> −2γn+ (1−δ)(1−φ(θ))

> ₋2γn+ (1−δ)λCαn−α.

TakingC large enough and usingγn=O(n−α)shows that (B.9) holds. This ends the proof of (B.6). Acknowledgements. The authors kindly thank M. Barlow, F. den Hollander and P. Mathieu for useful discussions.

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