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VELOCITY ESTIMATES FOR SYMMETRIC RANDOM WALKS AT LOW BALLISTIC DISORDER CLÉMENT LAURENT,ALEJANDROF. RAMÍREZ,CHRISTOPHESABOT ANDSANTIAGOSAGLIETTI Abstract. We derive asymptotic estimates for the velocity of random walks in random environments which are perturbations of the simple symmetric random walk but have a small local drift in a given direction. Our estimates complement previous results presented by Sznitman in [Sz03] and are in the spirit of expansions obtained bySabot in [Sa04]. 7 1 1. Introduction and Main Results 0 2 The mathematical derivation of explicit formulas for fundamental quantities of the model of n random walk inarandom environment isachallenging problem. For quantities likethevelocity, the a variance or the invariant measure of the environment seen from the random walk, few results exist J (seeforexamplethereview[ST16]forthecaseofDirichletenvironments, [DR14]forone-dimensional 3 computations and also [Sa04, CR16] for multidimensional expansions). In [Sa04], Sabot derived an 2 asymptotic expansion for the velocity of the random walk at low disorder under the condition that ] the local drift of the perturbed random walk is linear in the perturbation parameter. As a corollary R one candeduce that, inthecaseofperturbations of thesimple symmetricrandom walk, thevelocity P is equal to the local drift with an error which is cubic in the perturbation parameter. In this article . h we explore up to which extent this expansion can be generalized to perturbations which are not t a necessarily linear in the perturbation parameter and we exhibit connections with previous results m of Sznitman about ballistic behavior [Sz03]. [ Fix an integer d 2 and for x = (x ,...,x ) Zd let x := x + + x denote its l1-norm. 1 d 1 d ≥ ∈ | | | | ··· | | 1 Let V := x Zd : x = 1 be the set of canonical vectors in Zd and denote the set of all 1 v probability{vec∈torsp~=| (|p(e)) } onV,i.e. suchthatp(e) 0forall e V aPnd also p(e) = 1. 8 e∈V ≥ ∈ e∈V 0 Furthermore, let us consider the product space Ω := Zd endowed with its Borel σ-algebra (Ω). P P B 3 We call any ω = (ω(x)) Ω an environment. Notice that, foreach x Zd, ω(x)is aprobability 6 x∈Zd ∈ ∈ vector on V, whose components we will denote by ω(x,e) for e V, i.e. ω(x) = (ω(x,e)) . 0 ∈ e∈V . The random walk in the environment ω starting from x Zd is then defined as the Markov chain 1 ∈ (X ) with state space Zd which starts from x and is given by the transition probabilities 0 n n∈N0 7 P (X = y+eX = y) = ω(y,e), 1 x,ω n+1 | n : for all y Zd and e V. We will denote its law by P . We assume throughout that the space of v x,ω ∈ ∈ i environments Ω is endowed with aprobability measure P, called the environmental law.We will call X P the quenched law of the random walk, and also refer to the semi-direct product P := P P x,ω x x,ω r ⊗ a defined on Ω ZN as the averaged or annealed law of the random walk. In general, we will call × the sequence (X ) under the annealed law a random walk in a random environment (RWRE) n n∈N0 with environmental law P. Throughout the sequel, we will always assume that the random vectors Date: January 24, 2017. 2010 Mathematics Subject Classification. 60K37, 82D30, 82C41. Key words and phrases. Random walk in random environment,Green function, asymptotic expansion. AlejandroRamírezandSantiagoSagliettihavebeenpartiallysupportedbyIniciativaCientíficaMilenioNC120062 andbyFondoNacionaldeDesarrolloCientíficoyTecnológicogrant1141094. ClémentLaurenthasbeenpartiallysup- portedbyFondoNacionaldeDesarrolloCientíficoyTecnológicopostdoctoralgrant3130353. AlejandroRamirezand Christophe Sabot havebeen partially suported by MathAmsud project “Large scale behavior of stochastic systems”. 1 VELOCITY ESTIMATES FOR RWRE 2 (ω(x)) are i.i.d. under P. Furthermore, we shall also assume that P is uniformly elliptic, i.e. x∈Zd that there exits a constant κ > 0 such that for all x Zd and e V one has ∈ ∈ P(ω(x,e) κ) = 1. ≥ Given l Sd−1, we will say that our random walk (X ) is transient in direction l if ∈ n n∈N0 lim X l = + P a.s., n 0 n→∞ · ∞ − and say that it is ballistic in direction l if it satisfies the stronger condition X l n liminf · > 0 P a.s. 0 n→∞ n − Any random walk which is ballistic with respect to some direction l satisfies a law of large numbers (see [DR14] for a proof of this fact), i.e. there exists a deterministic vector ~v Rd with ~v l > 0 ∈ · such that X n lim =~v P a.s.. 0 n→+∞ n − This vector ~v is known as the velocity of the random walk. Throughoutthefollowingwewillfixacertaindirection, saye := (1,0,...,0) Sd−1 forexample, 1 ∈ and study transience/ballisticity only in this fixed direction. Thus, whenever we speak of transience or ballisticity of the RWRE it will be understood that it is with respect to this given direction e . 1 However, we point out that all of our results can be adapted and still hold for any other direction. For our main results, we will consider environmental laws P which are small perturbations of the simple symmetric random walk. More precisely, we will work with environmental laws P supported on the subset Ω Ω for ǫ > 0 sufficiently small, where ǫ ⊆ 1 ǫ Ω := ω Ω: ω(x,e) for all x Zd and e V . (1) ǫ ∈ − 2d ≤ 4d ∈ ∈ (cid:26) (cid:12) (cid:12) (cid:27) (cid:12) (cid:12) Notice that if P is supported on Ω for some ǫ 1 then it is uniformly elliptic with constant (cid:12)ǫ (cid:12) (cid:12) (cid:12)≤ 1 κ = . (2) 4d Since we wish to focus on RWREs for which there is ballisticity in direction e , it will be necessary 1 to impose some further conditions on the environmental law P. Indeed, if for each x Zd we define ∈ the local drift of the RWRE at site x as the random vector d~(x) := ω(x,e)e e∈V X then, in order for the walk to be ballistic in direction e , one could expect that it is enough to have 1 λ := E(d~(0)) e > 0, where E here denotes the expectation with respect to the law P (notice that 1 · all local drift vectors (d~(x)) are i.i.d. so that it suffices to consider only the local drift at 0). x∈Zd However, as shown in[BSZ03], there are examples of environments forwhich there existsadirection in which the expectation of the local drift is positive but the velocity of the corresponding RWREis negative. Therefore,wewillneedtoimposestrongerconditionsonthelocaldriftto have ballisticity, specifyingexactlyhowsmallweallowλtobe. Inthesequel,wewillconsidertwodifferentconditions, the first of which is quadratic local drift condition. Quadratic local drift condition (QLD). Given ǫ (0,1), we say that the environmental law P ∈ satisfies the quadratic local drift condition (QLD) if P(Ω ) = 1 and, furthermore, ǫ ǫ λ := E(d~(0)) e ǫ2. 1 · ≥ Our second condition, the local drift condition, is weaker for dimensions d 3. ≥ Local drift condition (LD). Given η,ǫ (0,1), we say that an environmental law P satisfies the ∈ local drift condition (LD) if P(Ω ) = 1 and, furthermore, η,ǫ ǫ VELOCITY ESTIMATES FOR RWRE 3 λ := E(d~(0)) e ǫα(d)−η, (3) 1 · ≥ where 2 if d= 2 α(d) := 2.5 if d= 3 (4)  3 if d 4. ≥ Observe that for d = 2 and any ǫ (0,1)condition (LD)η,ǫ implies (QLD)ǫ for all η (0,1), ∈ ∈ whereas if d 3 and η (0, 1) it is the other way round, (QLD) implies (LD) . It is known that ≥ ∈ 2 ǫ η,ǫ for every η (0,1) there exists ǫ = ǫ(d,η) > 0 such that any RWRE with an environmental law P 0 ∈ satisfying (LD) for some ǫ (0,ǫ ) is ballistic. Indeed, for d 3 this was proved by Sznitman in η,ǫ 0 ∈ ≥ [Sz03] whereas the case d = 2 was shown in [R16] (and is also a consequence of Theorem 2 below). Therefore, any RWRE with an environmental law P which satisfies (LD) for ǫ sufficiently small η,ǫ is such that P -a.s. the limit 0 X n ~v := lim n→∞ n exists and is different from 0. Our first result is then the following. Theorem 1. Given any η (0,1) and δ (0,η) there exists some ǫ = ǫ (d,η,δ) (0,1) such that, 0 0 ∈ ∈ ∈ for every ǫ (0,ǫ ) and any environmental law satisfying (LD) , the associated RWRE is ballistic 0 η,ǫ ∈ with a velocity ~v which verifies 0 <~v e λ+c ǫα(d)−δ (5) 1 0 · ≤ for some constant c = c (d,η,δ) > 0. We abbreviate (5) by writing 0<~v e λ+O (ǫα(d)−δ). 0 0 1 d,η,δ · ≤ Our second result is concerned with RWREs with an environmental law satisfying (QLD). Theorem 2. There exists ǫ (0,1) depending only on the dimension d such that for all ǫ (0,ǫ ) 0 0 ∈ ∈ and any environmental law satisfying (QLD) , the associated RWRE is ballistic with a velocity ~v ǫ which verifies ǫ2 ~v e λ . 1 | · − | ≤ d Combining both results we immediately obtain the following corollary. Corollary 3. Given δ (0,1) there exists some ǫ = ǫ (d,δ) (0,1) such that, for all ǫ (0,ǫ ) 0 0 0 ∈ ∈ ∈ and any environmental law satisfying (QLD) , the associated RWRE is ballistic with a velocity ~v ǫ which verifies ǫ2 λ ~v e λ+O (ǫα(d)−δ). 1 d,δ − d ≤ · ≤ Observe that for dimension d = 2 all the information given by Theorem 1 and Corollary 3 is already contained in Theorem 2, whereas this is not so for dimensions d 3. To understand better ≥ the meaning of our results, let us give some background. First, for x Zd and e V let us rewrite ∈ ∈ our weights ω(x,e) as 1 ω(x,e) = +ǫξ (x,e), (6) ǫ 2d where 1 1 ξ (x,e) := ω(x,e) . ǫ ǫ − 2d (cid:18) (cid:19) Notice that if P(Ω ) = 1 then P-almost surely we have ξ (x,e) 1 for all x Zd and e V. ǫ | ǫ | ≤ 4d ∈ ∈ In [Sa04], Sabot considers a fixed environment p Ω together with an i.i.d. sequence of bounded 0 ∈ VELOCITY ESTIMATES FOR RWRE 4 randomvectorsξ = (ξ(x)) [ 1,1]V whereeachξ(x) = (ξ(x,e)) satisfies ξ(x,e) = 0. x∈Zd ⊆ − e∈V e∈V Then, he defines for each ǫ > 0 the random environment ω on any x Zd and e V as ∈ ∈P ω(x,e) := p (e)+ǫξ(x,e). 0 In the notation of (6), this corresponds to choosing p (e) = 1 and ξ (x,e) := ξ(x,e) not depending 0 2d ǫ onǫ. Undertheassumption thatthelocaldriftassociatedtothis RWREdoesnotvanish,itsatisfies Kalikow’s condition [K81]and thus ithas anon-zero velocity~v. Sabotthenproves thatthis velocity satisfies the following expansion: for any small δ > 0 there exists some ǫ = ǫ (d,δ) > 0 such that 0 0 for any ǫ (0,ǫ ) one has that 0 ∈ ~v = d~ +ǫd~ +ǫ2d~ +O ǫ3−δ , (7) 0 1 2 d,δ where (cid:16) (cid:17) d~ := p (e)e, d~ := E[ξ(0,e)]e, 0 0 1 e∈V e∈V X X and d~2 := Ce,e′Je′ e, ! e∈V e′∈V X X with Ce,e′ := Cov(ξ(0,e),ξ(0,e′)) and Je := gp0(e,0)−gp0(0,0). Hereg (x,y) denotes the Green’s function ofarandom walk with jumpkernel p . Itturns out that p0 0 for the particular case in which p is the jump kernel of a simple symmetric random walk (which is 0 the choice we make in this article), we have that d~ =0 and also d~ = 0. In particular, for this case 0 2 we have λ = ǫd~ e = O(ǫ) and 1 1 · ~v e = λ+O ǫ3−δ . (8) 1 d,δ · Eventhoughthisexpansionwasonlyshownvalidinthe(cid:16)regim(cid:17)eλ = O(ǫ),fromitonecanguessthat, at least at a formal level, the random walk should be ballistic whenever λ ǫ3−η for any η > δ. ≥ This was established previously by Sznitman from [Sz03] for dimensions d 4, but remains open ≥ for dimensions d = 2 and d = 3. In this context, our results show that under the drift condition (LD), which is always weaker than the λ = O(ǫ) assumption in [Sa04], for d = 2 the random walk is indeed ballistic and the expansion (8) is still valid up to the second order (Theorem 2), whereas for d 3 we show that at least an upper estimate compatible with the right-hand side of (8) holds ≥ for the velocity (Theorem 1). The proof of Theorem 1 is rather different from the proof of the velocity expansion (7) of [Sa04], and is based on a mixture of renormalization methods together with Green’s functions estimates, inspiredinmethodspresentedin[Sz03,BDR14]. Asafirststep,oneshowsthattheaveraged velocity of the random walk at distances of order ǫ−4 is precisely equal to the average of the local drift with anerroroforderǫα(d)−δ. Todothis, essentially weshowthatarightapproximation forthebehavior of the random walk at distances ǫ−1 is that of a simple symmetric random walk, so that one has to find a good estimate for the probability to move to the left or to the right of a rescaled random walk moving on a grid of size ǫ−1. This last estimate is obtained through a careful approximation of the Green’s function of the random walk, which involves comparing it with its average by using a martingale method. This is a crucial step which explains the fact that one loses precision in the error of the velocity in dimensions d = 2 and d = 3 compared with d 4. As a final result of ≥ these computations, we obtain that the polynomial condition of [BDR14] holds. In the second step, we use a renormalization method to derive the upper bound for the velocity, using the polynomial condition proved in the first step as a seed estimate. The proof of Theorem 2 is somewhat simpler, and is based on a generalization of Kalikow’s formula proved in [Sa04] and a careful application of Kalikow’s criteria for ballisticity. VELOCITY ESTIMATES FOR RWRE 5 The article is organized as follows. In Section 2 we introduce the general notation and establish some preliminary facts about the RWRE model, including some useful Green’s function estimates. In Section 3, we prove Theorem 2. In Section 4, we obtain the velocity estimates for distances of order ǫ−4 which is the first step in the proof of Theorem 1. Finally, in Section 5 we finish the proof of Theorem 1 through the renormalization argument described above. 2. Preliminaries Inthissectionweintroducethegeneralnotationtobeusedthroughoutthearticleandalsoreview some basic facts about RWREs which we shall need later. 2.1. General notation. Given any subset A Zd, we define its (outer) boundary as ⊂ ∂A:= x Zd A : x y = 1 for some y A . { ∈ − | − | ∈ } Also, we define the first exit time of the random walk from A as T := inf n 0: X / A . A n { ≥ ∈ } In the particular case in which A = b Zd−1 for some b Z, we will write T instead of T , i.e. b A { }× ∈ T := inf n 0 :X e = b . b n 1 { ≥ · } Throughout the rest of this paper ǫ > 0 will be treated as a fixed variable. Also, we will denote generic constants by c ,c ,.... However, whenever we wish to highlight the dependence of any of 1 2 these constants on the dimension d or on η, we will write for example c (d) or c (η,d) instead of c . 1 1 1 Furthermore, for the sequel we will fix a constant θ (0,1) to be determined later and define ∈ L := 2[θǫ−1] (9) where [] denotes the (lower) integer part and also · N := L3, (10) which willbeusedaslengthquantifiers. Inthesequelwewilloftenworkwithslabsandboxes in Zd, which we introduce now. For each M N, x Zd and l Sd−1 we define the slab ∈ ∈ ∈ U (x) := y Zd : M (y x) l < M . (11) l,M ∈ − ≤ − · n o Whenever l = e we will suppress l fromthe notation and write U (x) instead. Similarly, whenever 1 M x = 0 we shall write U instead of U (0) and abbreviate U (0) simply as U for L as defined (9). M M L Also, for each M N and x Zd, we define the box ∈ ∈ M B (x) := y Zd : < (y x) e < M and (y x) e <25M3 for 2 i d (12) M 1 i ∈ − 2 − · | − · | ≤ ≤ (cid:26) (cid:27) together with its frontal side ∂+BM(x) := y ∂BM,M′(x) : (y x) e1 M , ∈ − · ≥ its back side (cid:8) (cid:9) M ∂−BM(x) := y ∂BM,M′(x) :(y x) e1 , ∈ − · ≤ − 2 (cid:26) (cid:27) its lateral side ∂lBM(x) := y ∂BM,M′(x) : (y x) ei 25M3 for some 2 i d , ∈ | − · |≥ ≤ ≤ and, finally, its middle-fro(cid:8)ntal part (cid:9) M B∗ (x) := y B (x) : (y x) e < M, (y x) e < M3 for 2 i d M ∈ M 2 ≤ − · 1 | − · i| ≤ ≤ (cid:26) (cid:27) VELOCITY ESTIMATES FOR RWRE 6 together with its corresponding back side M ∂ B∗ (x) := y B∗ (x) : (y x) e = . − M ∈ M − · 1 2 (cid:26) (cid:27) As inthecaseofslabs, we willusethesimplified notation B := B (0)and also∂ B := ∂ B (0) M M i M i M for i= +, ,l, with the analogous simplifications for B∗ (0) and its back side. − M 2.2. Ballisticity conditions. For the development of the proof of our results, it will be important to recall a few ballisticity conditions, namely, Sznitman’s (T) and (T′) conditions introduced in [Szn01,Szn02]andalsothepolynomialconditionpresentedin[BDR14]. Wedothisnow,considering only ballisticity in direction e for simplicity. 1 Conditions (T) and (T′). Given γ (0,1] we say that condition (T) is satisfied (in direction e ) γ 1 ∈ if there exists a neighborhood V of e in Sd−1 such that for every l′ V one has that 1 ∈ 1 limsup logP X l′ <0 < 0. (13) M→+∞ Mγ 0 TUl′,M · (cid:16) (cid:17) As a matter of fact, Sznitman originally introduced a condition (T) which is slightly different from γ the one presented here, involving an asymmetric version of the slab Ul′,M in (13) and an additional parameter b > 0 which modulates the asymmetry of this slab. However, it is straightforward to check that Sznitman’s original definition is equivalent to ours, so we omit it for simplicity. Having defined the conditions (T) for all γ (0,1], we will say that: γ ∈ (T) is satisfied (in direction e ) if (T) holds, 1 1 • (T′) is satisfied (in direction e ) if (T) holds for all γ (0,1). 1 γ • ∈ It is clear that (T) implies (T′), although it is not yet known whether the other implication holds. Condition (P) . Given K N we say that the polynomial condition (P) holds (in direction e ) K K 1 ∈ if for some M M one has that 0 ≥ 1 sup P X / ∂ B , x∈B∗ x TBM ∈ + M ≤ MK M (cid:16) (cid:17) where M := exp 100+4d(logκ)2 (14) 0 whereκistheuniformellipticity constant, whi(cid:8)chinourpresent(cid:9)casecanbetakenasκ= 1 , see (2). 4d It is well-known that both (T′) and (P) imply ballisticity in direction e , see [Szn02, BDR14]. K 1 Furthermore, in [BDR14] it is shown that (P) holds for some K 15d+5 (T′) holds (T) holds for some γ (0,1). K γ ≥ ⇐⇒ ⇐⇒ ∈ 2.3. Green’s functions and operators. Let us now introduce some notation we shall use related to the Green’s functions of the RWRE and of the simple symmetric random walk (SSRW). Given a subset B Zd, the Green’s functions of the RWRE and SSRW killed upon exiting B ⊆ are respectively defined for x,y B ∂B as ∈ ∪ TB g (x,y,ω) := E and g (x,y) := g (x,y,ω ), B x,ω 1{Xn=y}! 0,B B 0 n=0 X where ω is the corresponding weight of the SSRW, given for all x Zd and e V by 0 ∈ ∈ 1 ω (x,e) = . 0 2d VELOCITY ESTIMATES FOR RWRE 7 Furthermore, if ω Ω is such that E (T ) < + for all x B, we can define the corresponding x,ω B ∈ ∞ ∈ Green’s operator on L∞(B) by the formula G [f](x,ω) := g (x,y,ω)f(y). B B y∈B X Noticethatg ,andthereforealsoG ,dependson ω only thoughitsrestriction ω toB. Finally, it B B B | isstraightforward tocheckthatifB isaslabasdefinedin(11)thenbothg andG arewell-defined B B for all environments ω Ω with ǫ (0,1). ǫ ∈ ∈ 3. Proof of Theorem 2 The proof of Theorem 2 has several steps. We begin by establishing a law of large numbers for the sequence of hitting times (T ) . n n∈N 3.1. Law of large numbers for hitting times. We now show that, under the condition (P) , K thesequenceofhittingtimes (T ) satisfiesalawoflargenumbers withtheinverseofthevelocity n n∈N in direction e as its limit. 1 Proposition 4. If (P) is satisfied for some K 15d+5 then P -a.s. we have that K 0 ≥ E (T ) T 1 0 n n lim = lim = > 0, (15) n→∞ n n→∞ n ~v e1 · where ~v is the velocity of the corresponding RWRE. To prove Proposition 4, we will require the following lemma and its subsequent corollary. Lemma 5. If (P) holds for some K 15d+5 then there exists c > 0 such that for each n N K 1 ≥ ∈ and all a > 1 one has that v·e1 2d−1 Tn 1 1 2 2d−1 P0 a exp c1 log a +(log(n)) 2 . (16) n ≥ ≤ c − − ~v e (cid:18) (cid:19) 1 ( (cid:18) (cid:18) · 1(cid:19)(cid:19) !) Proof. By Berger, Drewitz and Ramírez [BDR14], we know that since (P) holds for K 15d+5, K ≥ necessarily (T′) must also hold. Now, a careful examination of the proof of Theorem 3.4 in [Szn02] shows that the upper bound in (16) is satisfied. (cid:3) Corollary 6. If (P) holds for some K 15d+5 then Tn is uniformly P -integrable. K ≥ n n∈N 0 Proof. Note that, by Lemma 5, for K > 1 and n 2 w(cid:0)e ha(cid:1)ve that v·e1 ≥ TndP0 ∞ (k+1)P0 Tn k 1 ∞ (k+1)e−c1(cid:16)log(cid:16)k−~v·1e1(cid:17)(cid:17)2d2−1−c1(log(2))2d2−1. Z{Tnn≥K} n ≤ kX=K (cid:18) n ≥ (cid:19) ≤ c1 kX=K From here it is clear that, since d 2, we have ≥ T n lim sup dP = 0 0 K→∞"n≥1Z{Tnn≥K} n # which shows the uniform P -integrability. (cid:3) 0 Let us now see how to obtain Proposition 4 from Corollary 6. Since (P) holds for K 15d+5, K ≥ by Berger, Drewitz and Ramírez [BDR14] we know that the position of the random walk satisfies a VELOCITY ESTIMATES FOR RWRE 8 law of large numbers with a velocity ~v such that ~v e > 0. Now, note that for any ε > 0 one has 1 · n X e P ~v e ε =P Tn · 1 ~v e ε 0 1 0 1 T − · ≥ T − · ≥ (cid:18)(cid:12) n (cid:12) (cid:19) (cid:18)(cid:12) n (cid:12) (cid:19) (cid:12) (cid:12) ∞ (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Xk e1 (cid:12) (cid:12) (cid:12) P(cid:12) · ~v (cid:12)e ε 0 1 ≤ k − · ≥ k=n (cid:18)(cid:12) (cid:12) (cid:19) X (cid:12) (cid:12) ∞ (cid:12) 2d−1 (cid:12) e−C(l(cid:12)ogk) 2 , (cid:12) ≤ k=n X where in the last inequality we have used the slowdown estimates for RWREs satisfying (T′) proved by Sznitman in [Szn02] (see also the improved result of Berger in [B12]). Hence, by Borel-Cantelli we conclude that P -a.s. 0 n lim =~v e , 1 n→∞Tn · fromwherethesecondequalityof (15)immediatelyfollows. Thefirstoneisnowadirectconsequence of the uniform integrability provided by Corollary 6. 3.2. Introducing Kalikow’s walk. Given a nonempty connected strict subset B ( Zd, for x B ∈ we define Kalikow’s walk on B (starting from x) as the random walk starting from x which is killed upon exiting B and has transition probabilities determined by the environment ω B given by B ∈ P E(g (x,y,ω)ω(y,e)) ωx(y,e) := B . (17) B E(g (x,y,ω)) B It is straightforward to check that by the uniform ellipticity of P we have 0 < E(g (x,y,ω)) < + B ∞ for all y B, so that the environment ωx is well-defined. In accordance with our present notation, ∈ B we will denote the law of Kalikow’s walk on B by Px,ωBx and its Green’s function by gB(x,·,ωBx). The importance of Kalikow’s walk, named after S. Kalikow who originally introduced it in [K81], lies in the following result which is a slight generalization of Kalikow’s formula proved in [K81] and of the statement of it given in [Sa04]. Proposition 7. If B ( Zd is connected then for any x B with Px,ωx(TB < + )= 1 we have ∈ B ∞ E(g (x,y)) = g (x,y,ωx) (18) B B B for all y B ∂B. ∈ ∪ Proof. The proof is similar to that of [Sa04, Proposition 1], but we include it here for completeness. First, let us observe that for any ω Ω and y B ∂B we have by the Markov property ǫ ∈ ∈ ∪ TB g (x,y,ω) = E B x,ω 1{Xn=y}! n=0 X ∞ = P (X = y,T n) x,ω n B ≥ n=0 X ∞ = (y)+ P (X =y e,X = y,T > n 1) {x} x,ω n−1 n B 1 − − n=1e∈V XX ∞ = (y)+ P (X =y e,T > n 1)ω(y e,e) {x} x,ω n−1 B 1 − − − e∈V n=1 XX = (y)+ (y e)g (x,y e,ω)ω(y e,e), {x} B B 1 1 − − − e∈V X VELOCITY ESTIMATES FOR RWRE 9 so that E(g (x,y)) = (y)+ E(g (x,y e))ωx(y e,e). B 1{x} B − B − e∈V :y−e∈B X Similarly, if for each k N we define 0 ∈ TB∧k gB(k)(x,y,ωBx):= Ex,ωBx 1{Xn=y}! n=0 X then by the same reasoning as above we obtain g(k+1)(x,y,ωx) = (y)+ g(k)(x,y e,ωx)ωx(y e,e). (19) B B 1{x} B − B B − e∈V :y−e∈B X In particular, we see that for all k N 0 ∈ E(g (x,y)) g(k+1)(x,y,ωx) = E(g (x,y e)) g(k)(x,y e,ωx) ωx(y e,e) B − B B B − − B − B B − e∈VX:y−e∈B(cid:16) (cid:17) which, since ωx isnonnegative andalsog(0)(x,y,ωx) = (y) E(g (x,y))forevery y B ∂B, B B B 1{x} ≤ B ∈ ∪ by induction implies that g(k)(x,y,ωx) E(g (x,y)) for all k N . Therefore, by letting k + B B ≤ B ∈ 0 → ∞ in this last inequality we obtain g (x,y,ωx) E(g (x,y)) (20) B B ≤ B for all y B ∂B. In particular, this implies that ∈ ∪ Px,ωBx(TB < +∞)= gB(x,y,ωBx)≤ E(gB(x,y)) = Px(TB < +∞)≤ 1. (21) y∈∂B y∈∂B X X Thus, if Px,ωx(TB < + ) = 1 then both sums on (21) are in fact equal which, together with (20), B ∞ implies that g (x,y,ωx) = E(g (x,y)) B B B for all y ∂B. Finally, to check that this equality also holds for every y B, we first notice that ∈ ∈ for any y B ∂B we have by (19) that ∈ ∪ g (x,y,ωx) = (y)+ g (x,y e,ωx)ωx(y e,e) B B 1{x} B − B B − e∈V :y−e∈B X so that if y B ∂B is such that E(g (x,y)) = g (x,y,ωx) then ∈ ∪ B B B 0= (E(g (x,y e)) g (x,y e,ωx))ωx(y e,e). B − − B − B B − e∈V :y−e∈B X Hence, by the nonnegativity of ωx and (20) we conclude that if y B ∂B is such that (18) holds B ∈ ∪ then (18) also holds for all z B of the form z = y e for some e V. Since we already have that (18) holds for all y ∂B and∈B is connected, by ind−uction one can∈obtain (18) for all y B. (cid:3) ∈ ∈ As a consequence of this result, we obtain the following useful corollary, which is the original formulation of Kalikow’s formula [K81]. Corollary 8. If B ( Zd is connected then for any x B such that Px,ωx(TB < + )= 1 we have ∈ B ∞ Ex(TB) = Ex,ωx(TB) B and Px(XTB = y) = Px,ωBx(XTB = y) for all y ∂B. ∈ VELOCITY ESTIMATES FOR RWRE 10 Proof. This follows immediately fromProposition 7uponnoticing that, by definition of g , we have B on the one hand Ex(TB)= E(gB(0,y)) = gB(x,y,ωBx) = Ex,ωBx(TB) y∈B y∈B X X and, on the other hand, for any y ∂B ∈ Px(XTB = y) =E(gB(x,y)) = gB(x,y,ωBx) = Px,ωBx(XTB = y). (cid:3) Proposition 4 shows that in order to obtain bounds on~v e , the velocity in direction e , it might 1 1 · be useful to understand the behavior of the expectation E (T ) as n tends to infinity, provided that 0 n the polynomial condition (P) indeed holds for K sufficiently large. As it turns out, Corollary 8 K will provide a way in which to verify the polynomial condition together with the desired bounds for E (T ) by means of studying the killing times of certain auxiliary Kalikow’s walks. To this end, 0 n the following lemma will play an important role. Lemma 9. If given a connected subset B ( Zd and x B we define for each y B the drift at y ∈ ∈ of the Kalikow’s walk on B starting from x as d~ (y) := ωx(y,e)e B,x B e∈V X where ωx is the environment defined in (17), then B d~(y,ω) E d~ (y)= e∈V ω(x,e)fB,x(y,y+e,ω) . B,x (cid:16)P (cid:17) E 1 e∈V ω(x,x+e)fB,x(y,y+e,ω) (cid:16)P (cid:17) where f is given by B,x P (T H ) z,ω B y f (y,z,ω) := ≤ B,x P (H < T ) x,ω y B and H := inf n N : X = y denotes the hitting time of y. y 0 n { ∈ } Proof. Observe that if for y,z B ∂B and ω Ω we define ∈ ∪ ∈ g(y,z,ω) := P (H < T ) z,ω y B then by the strong Markov property we have for any y B ∈ TB TB E(g (x,y,ω)) = E E = E g(y,x,ω)E . B x,ω 1{Xn=y}!! y,ω 1{Xn=y}!! n=0 n=0 X X Now, under the law P , the total number of times n N in which the random walk X is at y y,ω 0 ∈ before exiting B is a geometric random variable with success probability p := ω(y,y+e)(1 g(y,y +e,ω)), − e∈V X so that TB 1 E = . y,ω n=01{Xn=y}! e∈V ω(y,y+e)(1−g(y,y+e,ω)) X It follows that P 1 E(g (x,y,ω)) = E B ω(y,y+e)f (y,y+e,ω) (cid:18) e∈V B,x (cid:19) P