PHD THESIS: Topics in Ballistic and Transient Conditions for Random Walks in Random Environments A Thesis submitted by Enrique Guerra Aguilar for the degree of Doctor en Matemáticas in the Pontificia Universidad Católica de Chile Supervised by: Alejandro F. Ramı́rez July, 2016 Santiago, Chile Dedicado a mi esposa Stephanie Alfaro y a mis padres Enrique Guerra Galaz y Patricia Aguilar Jara. i Acknowledgements I would here like to thank all people responsible for making this thesis more than a simple keyboard work. First of all, I express my sincere thanks to my thesis advisor Alejandro Ramı́rez for letting me to be his student. It was a real pleasure to work with him, I learnt much more than mathematics. He is undoubtedly my mentor and all my results were turned out by a deep exchange of probabilistic ideas. I also thank my thesis committee: Joaquin Fontbona, Gregorio Moreno, Rolando Rebolledo and Christophe Sabot. I offer my apologies for the delay in delivering the final thesis version. I have studied for more than a decade at the Pontificia Universidad Catolica de Chile. There I have met and interacted with many people. Among friends and staff whom I met, I would like to especially thank Dr. Moreno and Dr. Cabezas for sharing their knowledge through which often made me strengthen my own knowledge and like for the probability field. Also, I thank my office mate Alvaro Ferrada for his friendship, who is my friend from the very beginning of my graduate studies. I thank the partial support throughout my research to the Nucleo Milenio: Modelos Estocásticos de Sistemas Complejos y Desordenados, a mathematical community based at the Pontificia Universidad Católica and the Universidad de Chile. Regarding the Nucleo staff, for the constant help in non-mathematical matters I express my gratitude to: Consuelo Thiers, Maria Eugenia Heckman and Cecile Jourdan. Last but most important, I deeply thank with all of my heart my family: my wife Stephanie Alfaro who is the main source of inspiration in each mathematical idea that I have, and my parents: Enrique Guerra and Patricia Aguilar whom I owe all what I am. ii Abstract This thesis is devoted to the study of the stochastic process model called Random Walk in Random Environment (RWRE). To be precise, our research focuses on two kinds of random environments. The first one is the so called uniformly elliptic i.i.d. random environment. In this model it is conjectured that in dimensions d ě 2 any random walk which is directionally transient is ballistic. The ballisticity conditions for RWRE somehow interpolate between directional transience and ballisticity and have served to quantify the gap which one needs to answer affirmatively this conjecture. Two important ballisticity conditions introduced by Sznitman [Sz02] in 2001 and 2002 are the so called conditions pT 1 q and pT q: given a slab of width L orthogonal to l, condition pT 1 q in direction l is the requirement that the annealed exit probability of the walk through the side of the γ slab in the half-space tx : x ¨ l ă 0u, decays faster than e´CL for all γ P p0, 1q and some constant C ą 0, while condition pT q in direction l is the requirement that the decay is exponential e´CL . It is believed that pT 1 q implies pT q. We show that pT 1 q implies at least an almost (in a sense to be made precise) exponential decay. The second class of random environment to be studied is a larger class which only requires a mixing condition on the environment law. As a matter of fact, the ballisticity conditions in this framework are not well-understood. Therefore our purpose is to find a connection between this strictly larger class of environments and the ballisticity conditions which have proved to be a powerful theoretical concept for random walks in an i.i.d. random environment. In that direction, we prove that every random walk in a uniformly elliptic random environment satisfying the cone mixing condition and a non-effective polynomial ballisticity condition with high enough degree has an asymptotic direction. iii Contents Acknowledgements ii Abstract. iii List of Figures vi General Introduction 0.1 1 Some well-known results in uniformly elliptic i.i.d. random environments. (under d ě 2.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 0.1.1 On Kalikow’s Condition. . . . . . . . . . . . . . . . . . . . . . . . . 6 0.1.2 Ballisticity Conditions: Stretched Exponential Decay, Effective Criterion and Polynomial Condition . . . . . . . . . . . . . . . . . . . 9 0.2 Previous results for random walks in cone mixing random environments . . 13 0.3 A brief explanation of the Thesis Results . . . . . . . . . . . . . . . . . . . 17 0.3.1 Main Result for i.i.d. Random Environments . . . . . . . . . . . . . 17 0.3.2 Main Result for cone mixing random environments . . . . . . . . . 18 1 Almost exponential decay for the exit probability from slabs of ballistic RWRE 21 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.2 Proof of Theorem 1.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.2.1 Preliminaries and notation . . . . . . . . . . . . . . . . . . . . . . . 27 1.2.2 The maximal growth condition on scales . . . . . . . . . . . . . . . 29 1.2.3 An adequate choice of fast-growing scales . . . . . . . . . . . . . . . 32 1.2.4 The effective criterion implies Theorem 1.1.2 . . . . . . . . . . . . . 38 iv 2 Asymptotic Direction for Random Walk in Strong Mixing Environment 45 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.2 Preliminary discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.2.1 Non-effective polynomial condition and its relation with other directional transience conditions . . . . . . . . . . . . . . . . . . . . . 51 2.3 2.2.2 Cone mixing and ergodicity . . . . . . . . . . . . . . . . . . . . . . 54 2.2.3 Polynomial Decay implies Polynomial decay in a neighborhood . . . 56 Examples of directionally transient random walks without an asymptotic direction and vanishing velocity . . . . . . . . . . . . . . . . . . . . . . . . 61 2.3.1 Random walk with a vanishing velocity but with an asymptotic direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.3.2 Directionally transient random walk without an asymptotic direction 65 2.4 Backtracking of the random walk out of a cone . . . . . . . . . . . . . . . . 69 2.5 Polynomial control of regeneration positions . . . . . . . . . . . . . . . . . 77 2.6 2.7 2.5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.5.2 Preparatory results . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2.5.3 Proof of Proposition 2.5.3 . . . . . . . . . . . . . . . . . . . . . . . 87 Proof of Theorem 2.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.6.1 Approximate regeneration time sequence . . . . . . . . . . . . . . . 98 2.6.2 Approximate asymptotic direction . . . . . . . . . . . . . . . . . . . 102 2.6.3 Proof of Theorem 2.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . 105 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 v List of Figures 1 Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The choice of boxes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.2 A geometric sketch to bound Q2 rXTBL,2L,l p0q R B ` BL,2L,l p0qs. . . . . . . . . . 69 2.3 The boxes By and Bz are inside of Cp0, l, αq. . . . . . . . . . . . . . . . . . 85 vi 3 General Introduction Random Walk in a Random Environment (RWRE) is a classical model of random motion in a random media. It was originally introduced as a toy model for replication of DNA chains and phase transition in alloys. We can describe a d- dimensional RWRE as the canonical Markov chain pXn qně0 with state space Zd , where its transition probabilities to nearest neighbor sites are random. In spite of its simplicity, when the dimension is larger than 1 its asymptotic laws are still not well-understood. This problem, is essentially due to the reversibility loss in the chain on averaging over the environment. Consequently this makes it hard to apply standard convergence methods in order to get asymptotic laws. Some progress has been done in that direction by means of the introduction of what are called ballisticity conditions. These conditions are essentially a functional control (for instance a polynomial control) of the probability that the walk exits from large slabs transversal to directions l1 in a neighborhood of a given direction l P Sd´1 by the unlikely slab boundary side: the one which is in the direction ´l1 . The study of ballisticity conditions is the main focus of this thesis. To appropriately introduce them, we will now explain more precisely the model. Let d ě 1 be a positive integer which will be thought as the underlying random walk dimension. We consider the p2d ´ 1q- dimensional simplex P defined by: 2d P :“ tz P R : 2d ÿ zi “ 1, zi ě 0 for i P r1, 2dsu. i“1 d Now, an environment ω :“ ωpx, eq|xPZd ,ePZd ,|e|“1 is an element of the set Ω :“ pPqZ which specifies at each site x P Zd the transition probabilities of the walk. Throughout this chapter, by canonical σ- algebra on a product space we mean the σ-algebra generated by the cylinder measurable sets. For the time being, we assume that we have a given probability measure P on the canonical σ- algebra W in Ω. 1 For fixed ω P Ω and x P Zd , one defines the quenched law Px,ω , as the law of the canonical Markov chain pXn qně1 starting from x, with state space Zd and satisfying Px,ω rX0 “ xs “ 1 Px,ω rXn`1 “ Xn ` e | Xn , Xn´1 , . . . X0 s “ ωpXn , eq for e P Zd , |e| “ 1. We call F the canonical σ- algebra in pZd qN , which is the σ- algebra in the walk path space. Furthermore, for a prescribed probability measure P one then defines the annealed or averaged law Px as the semi-direct product P b Px,ω on W ˆ F. We consider two types of random environments: the first one will be the so-called i.i.d. random environment framework; the second one is a larger class satisfying a mixing 1 condition. We start by defining the i.i.d. random environment. Let κ P p0, 2d s and µ be a probability measure on P such that for each x P Zd , ωpx, ¨q distributes as µ, and µ is supported on the subset Pκ of P defined by: Pκ :“ tz P R2d : 2d ÿ zi “ 1, zi ě κ for i P r1, 2dsu. i“1 This last restriction on the support of the law µ is called uniform ellipticity assumption. The random environment is now an element of the measurable probability space d Ωκ :“ pPκ qZ which is endowed with canonical σ´ algebra Wκ and the product measure d P :“ µbZ . For the easy of notation, we shall drop κ when we talk about i.i.d random environments. Before introducing the cone mixing condition we weaken the uniform ellipticity assumption. We say that P is uniformly elliptic with respect to l, denoted by pU Eq|l, if the jump probabilities of the random walk are positive and larger than 2κ in those directions for which the projection on l is positive. In other words if Prωp0, eq ą 0s “ 1 for |e| “ 1 and if ” ı P min ωp0, eq ě 2κ “ 1, ePE where E :“ d ď tsgnpli qei u ´ t0u i“1 2 (1) and by convention sgnp0q “ 0. It will be convenient to define what is understood by a cone in this work. We let α be a small positive real number and R be a rotation such that Rpe1 q “ l. (2) To define the cone, it will be useful to consider for each i P r2, ds, the directions l`i “ l ` αRpei q |l ` αRpei q| and l´i “ l ´ αRpei q . |l ´ αRpei q| The cone Cpx, l, αq centered in x P Rd is defined as Cpx, l, αqq :“ d č ( z P Rd : pz ´ xq ¨ l`i ě 0, pz ´ xq ¨ l´i ě 0 . (3) i“2 The following picture shows a cone centered at x in the lattice Z2 β = arctan(α) C(x, l, α) l x Figure 1: Cone We are ready to state the cone mixing condition. Define the canonical shifts tθx : x P Zd u by θx ωpyq :“ ωpx ` yq for all ω P Ω and x, y P Zd . Let us first recall the concept of ergodic measure. We say that a probability measure P is stationary if for all x P Zd and A P W one has that Ppθx´1 Aq “ PpAq. We say that P is ergodic, if whenever A P W is such that A “ θx´1 A for all x P Zd , one has that PpAq “ 0 or that PpAq “ 1. Now, let φ : r0, 8q Ñ r0, 8q with limrÑ8 φprq “ 0. We say that a stationary probability measure P satisfies the cone mixing assumption with respect to α, l and φ, denoted 3 pCM qα,φ |l if for every pair of events A, B, where PpAq ą 0, A P σtωpz, ¨q; z ¨ l ď 0u, and B P σtωpz, ¨q; z P Cprl, l, αqu, it holds that ˇ ˇ ˇ PrA X Bs ˇ ˇ ˇ ď φpr|l|1 q. ´ PrBs ˇ PrAs ˇ (4) Thus, we can consider assumption pCM qα,φ |l as a restriction on the P- dependence. As it was mentioned in [CZ01], it is important to allow strictly positive angles β :“ arctanpαq. Otherwise, when β “ 0 and the cone mixing assumption is satisfied for each l P Sd´1 , then the measure P is actually a finite range dependence law (see [Ze1] and [B1]). Furthermore, whenever a probability measure P satisfies the cone mixing assumption, it is ergodic (this will be proved in Chapter 2). We will be dealing with three important asymptotic concepts: • We say that the walk is transient in the direction l, if ” P0 ı lim Xn ¨ l “ 8 “ 1. nÑ8 • We say that the walk is ballistic in the direction l if „ Xn ¨ l ą 0 “ 1. P0 lim inf nÑ8 n (5) (6) Moreover, in this case we will also say that the walk has ballistic behavior. • We say that a non-zero d- dimensional deterministic vector v̂ is an asymptotic direction for the walk if „ P0 Xn Ñ v̂ lim nÑ8 |Xn | (7) holds. It is straightforward to see that any RWRE which is ballistic in direction l is transient in the same direction. In the 1 dimensional case, the walk asymptotic behavior is wellunderstood, and the results come from Smith-Wilkinson [SW69], Solomon [So75] and Alili [Al99]. Define ρ :“ ωp0, ´e1 q . ωp0, e1 q We then have the following transience criteria: 4 Theorem 0.0.1 (Smith-Wilinson, Solomon, Alili). Suppose that P is ergodic and that Erlnpρqs is defined (possibly ˘8), then (i) Erln ρs ă 0 implies P0 -a.s. lim Xn “ 8. (ii) Erln ρs ą 0 implies P0 -a.s. lim Xn “ ´8. (iii) Erln ρs “ 0 implies P0 -a.s. ´8 “ lim inf Xn ă lim sup Xn “ 8. As an example of the possible atypical behavior of RWRE, Sinai [Sin82] considered random walks in random environments satisfying the case piiiq of the previous theorem together with 0 ă Erpln ρq2 s ă 8, proving that: the position Xn of the random walk takes on values of order log2 pnq. This is in contrast to the ordinary random walk typical ? asymptotic behavior of the random variable Xn which is of order n. We also have a ballisticity criteria as follows: Theorem 0.0.2 (Smith-Wilkinson, Solomon). Assume that P is i.i.d. and uniformly elliptic. One has that P0 -a.s. (i) For Erρs ă 1, v “ (ii) For 1 Erρ´1 s (iii) For 1 ă 1´Erρs 1`Erρs Xn n Ñ v, where ą 0. ď 1 ď Erρs, v “ 0. 1 , Erρ´1 s v“ 1´Erρ´1 s 1`Erρ´1 s ă 0. From Jensen inequality we can see that there exist random walks in i.i.d. random environments which are directionally transient with vanishing velocity. However in the higher dimensional case the last possibility is not expected as the following conjecture shows: Conjecture 0.0.3 (d ě 2.). Any d- dimensional RWRE which is uniform elliptic, i.i.d. and transient in direction l, is ballistic in direction l. As it was remarked above, this conjecture is not true when the dimension is 1. Informally, this conjecture says that traps are negligible when the dimension d ě 2, and we mean by traps finite though arbitrary large regions in Zd where the walk spends a long 5 time with relatively high probability. In this direction, an intermediate problem has been solved by Simenhaus [Si07] Theorem 0.0.4 (Simenhaus). Assume that a d- dimensional random walk in a uniform elliptic i.i.d. random environment is transient in a neighborhood of the direction l. Then, there exists an asymptotic direction v̂ for the random walk. The hypothesis of the previous theorem are actually equivalent. Indeed, the converse implication of Theorem 0.0.4 is a straightforward application of Kalikow’s 0-1 law [K81]. The proof of this theorem strongly makes use of the independent structure of the environment. We will give some further comments about this result in Section 0.3. We want now to introduce the so-called ballisticity conditions and summarize what is known. 0.1 Some well-known results in uniformly elliptic i.i.d. random environments. (under d ě 2.) We present some important results for the uniformly elliptic i.i.d. random environment setting. The first result that we would like to mention is a relatively old one and comes from Kalikow in [K81] (we refer to this article for a further discussion). 0.1.1 On Kalikow’s Condition. In order to enlighten the nature of this condition, we will need some definitions. For a given set U P Zd we define its boundary BU by: BU :“ ty P Zd ´ U : Dz P U, |y ´ z| “ 1u, and also define the first time of exit from the set U , which we denote TU via: TU “ inftn ě 0 : Xn R U u. Kalikow introduced a useful auxiliary Markov chain related to the original chain pXn qně0 . More precisely, let U be a connected strict subset of Zd with 0 P U , for x P U Y BU we define the Kalikow’s law Ppx,U as the law of the canonical Markov chain pXn qně0 (we keep 6 the same notation because this makes sense in view of (0.1.1)) starting from x with state space in U Y BU and stationary transition probabilities given by: $ řTU 1tXn “xu ωpx,eqs & E0 r n“0 , x P U, |e|1 “ 1, řTU E r 0 n“0 1tXn “xu s PpU px, x ` eq “ % 1, x P BU, e “ 0, where the above expectations are finite thanks to the uniform ellipticity assumption. The previously mentioned main connection between this auxiliary chain and the original one is given by: Theorem 0.1.1 (Kalikow). Assume Pp0,U rTU ă 8s. Then P0 rTU ă 8s and XTU has the same distribution under either Pp0,U or P0 . When d ě 2, Kalikow’s condition was the first condition used to prove asymptotic laws for RWRE. In the seminal result of [K81], Kalikow was able to prove directional transience under what is currently known as Kalikow’s criteria. This is a priori a stronger requirement than Kalikow’s condition. Before we define formally these concepts, we would like to heuristically explain what Kalikow’s condition is and explain the general reasoning behind proofs of asymptotic laws for the walk under such an assumption. Kalikow’s condition is essentially the existence of a positive local drift for the auxiliary Markov chains over all connected strictly subset U Ă Zd , with 0 P U . Standard arguments show that this implies ballistic behavior for the auxiliary Markov chains. We then transfer this ballistic behavior to the walk by means of (0.1.1) and some extra probabilistic arguments. Kalikow’s condition with respect to some fixed direction l P Sd´1 is the following requirement: Definition 0.1.2 (Kalikow’s Condition). There exists a non-random real number δ ą 0, so that: inf U,x ÿ PpU px, x ` eqe ¨ l ą δ e, |e|“1 holds, where the infimum runs over all connected finite strict subsets U P Zd such that 0 P U. As an example of what was mentioned in the previous paragraph, one can see that under this condition appealing to property (0.1.1) and Azuma’s inequality, the following important result is satisfied: 7 Theorem 0.1.3 (Kalikow). Assume Kalikow’s condition in direction l. Then P0 rlim Xn ¨ l “ 8s “ 1 . We refer to [Ze1] for further details about the proof of this theorem using the ideas outlined here. This result was considerably improved by Sznitman and Zerner through the introduction of a renewal structure which is a higher dimensional analog of the onedimensional theoretical construction introduced by Kesten in [Ke77], and which can be defined in directional transient case (see (5)). This renewal structure stems from a random time τ1 which can be thought as the first time that the walk reaches a record level with respect to direction l and after this time the walk does never backtrack. One can use the renewal structure to prove the equivalence between the requirement of the ballisticity definition given in (6) with the following a priori stronger assumption (see [DR14]): P0 a.s. one has that Xn ¨ l nÑ8 n lim (8) exists, is positive and constant. In this case, we can then define the velocity as v :“ lim nÑ8 Xn . n Furthermore, from standard subsequence methods, it can be seen that the right candidate for the velocity v is v :“ E0 rXτ1 | D “ 8s , E0 rτ1 | D “ 8s (9) where D is the hitting time of the half space tz P Rd , z ¨ l ă 0u (c.f. (10)). Therefore a natural question is the following one: what kind of local condition on the environment does allow us to have a finite first moment for the random variable τ1 ? In that direction, by means of a clever use of Kalikow’s condition (see [SZ99]), Sznitman and Zerner proved that: E0 rτ1 s ă 8. As a result, in view of (9) we obtain the following: 8 Theorem 0.1.4 (Sznitman and Zerner). Under Kalikow’s condition with respect to direction l there exists a deterministic v P Rd , such that P0 - almost surely: Xn Ñ v n Moreover, one has that v ¨ l ą 0. In a subsequent article [Sz01], Sznitman was able to prove a Central Limit Theorem under Kalikow’s condition. However we are mostly interested here in the ballistic conditions pT γ q|l for γ P p0, 1s which were defined in [Sz03]. 0.1.2 Ballisticity Conditions: Stretched Exponential Decay, Effective Criterion and Polynomial Condition As in the previous section, we begin with some definitions. We define for a P R, the stopping times Tal and Tral with respect to the canonical filtration of the walk by: Tal :“ inftn ě 0 : Xn ¨ l ě au, together with Tral :“ inftn ě 0 : Xn ¨ l ď au. It will also be convenient to define the stopping time D as the first time that the walk hits the random half-space tz P Zd : pz ´ X0 q ¨ l ă 0u: D :“ inftn ě 0 : Xn ¨ l ă X0 ¨ lu. (10) The underlying rough thought in the renewal structure is the following: under transience in direction l, P0 ´ a.s. there should exist a finite random time τ1 such that Xτ1 is a record level in direction l, and after time τ1 the walk never backtracks. In [SZ99] the authors prove that transience in direction l is equivalent to P0 rτ1 ă 8s “ 1. On the other hand, suppose that the walk is transient in direction l. For large L we consider the slab AL,l defined by AL,l :“ tz : |z ¨ l| ď Lu. 9 Elementary probabilistic arguments let us conclude that ” ı ” ı l P0 XTAL,l ¨ l ă 0 “ P0 Tr´L ă TrLl Ñ 0 as L goes to infinity. The ballisticity conditions introduced by Sznitman are stretched exponential controls for the above probabilities. Definition 0.1.5 (Stretched Exponential Decay). Let γ P p0, 1s. We say that pT γ q|l holds, if ı ” 1 1 lim sup L´γ ln P0 Tr´l rbL ă TLl ă 0, (11) LÑ8 1 for rb ą 0, and each l1 in some neighborhood of l. We also say that pT q|l is fulfilled if pT γ q|l holds for each γ P p0, 1q, and we use for short the notation pT q|l :“ pT 1 q|l. Let us remark that we can get rid in the previous definition the constant rb. This can be proved using the strategies developed in the proofs of Proposition 2.2.3 and Lemma 2.5.5. From the definition, it is straightforward to see that for prescribed γ1 , γ2 P p0, 1q, with γ1 ă γ2 the following chain of implications holds: 1 pT q|l Ñ pT q|l Ñ pT γ2 q|l Ñ pT γ1 q|l. We actually expect even more: it is believed that all these conditions are equivalent. A non-negligible progress has been made regarding this conjecture. As a first step to address this question Sznitman proved in [Sz03], the implication: 1 pT γ q|l Ñ pT q|l, (12) for any γ ą 1{2. The tool used to prove this, is what is called the effective criterion, which is a higher dimensional version of standard ballisticity conditions for one-dimensional RWRE. In turn, it can be seen as the triggering condition in an induction probabilistic procedure. Its definition is a bit technical. Nevertheless, given its importance, we recall it here. Let l P Sd´1 be fixed and let R be a rotation on Rd so that Rpe1 q “ l. Let L ą 2 and L̃ ą 0. Consider the box Bl,L,L̃ :“ Rpp´pL ´ 2q, L ` 2q ˆ p´L̃, L̃qd´1 q X Zd , 10 with the positive part of its boundary B ` Bl,L,L̃ defined via B ` Bl,L,L̃ :“ BB X tx P Zd , x ¨ l ě L ` 2, |Rpei q ¨ x| ă L̃, i ě 2u. We also attach three random variables p, q and ρ to this box , defined by the following relations qpωq :“ P0,ω rXTB R B ` Bl,L,L̃ s “ 1 ´ ppωq l,L,L̃ along with ρ“ qpωq ppωq We are ready to define the effective criterion as follows: Definition 0.1.6 (Effective Criterion). Let l P Sd . Then the effective criterion with ? respect to l is satisfied if for some L ą c2 , L̃ P r3 d, L3 q and a P r0, 1s the requirement + # ˆ ˆ ˙˙ 3pd´1q 1 L̃d´1 L3d´2 Erρa s ă 1, c3 ln κ Here, c2 and c3 are dimension dependent constants. Notice that the effective criterion shares some similarities with the Solomon criterion Erρs ă 1 which ensures ballistic regime, as it can be seen from (0.0.2). We can be more precise yet with the statement of the equivalence in (12). Indeed the following theorem due to Sznitman uses the effective criterion as a pivotal condition. Theorem 0.1.7 (Sznitman). The following statements are equivalent: • Effective Criterion with respect to l. 1 • pT q|l . • pT γ q|l for 1 ą γ ą 1{2. The proof of this theorem can be found in [Sz03]. Furthermore, in [BDR14] N. Berger, A. Drewitz and A. F. Ramı́rez proved the equivalence between this criterion and a polynomial decay of the probability entering in (11). Specifically, the polynomial condition in 11 a non-effective form is the following requirement: let M ą 0. The polynomial condition pP ˚ qM |l is satisfied if: ´M lim L LÑ8 ı ” 1 l1 l r P0 T´rbL ă TL “ 0 for each l1 in some neighborhood of l and rb ą 0. One has the following: Theorem 0.1.8 (Berger, Drewitz and Ramı́rez). Suppose that pP ˚ qM |l is satisfied for some M ą 15d ` 5. Then the Effective Criterion with respect to l is satisfied. Let us remark that actually in [BDR14], an effective version of the above polynomial condition on boxes was introduced. This means that it is a condition that in principle can be verified looking at the environment in large but finite boxes. The authors in this article also proved that the Effective Criterion of Sznitman is implied by their polynomial effective condition. Thus, using this polynomial effective condition one can avoid the use of the Effective Criterion to check ballisticity. On the other hand, using the equivalence 1 between the Effective Criterion in direction l and pT q|l we conclude that Theorem 0.1.9 (Berger, Drewitz, Ramı́rez and Sznitman). The following conditions are equivalent: 1 • pT q|l. • pT γ q|l for 1 ą γ ą 0. • pP ˚ qM |l for M ą 15d ` 5. • Effective Criterion with respect to l. As it was tacitly induced in the name given to pT γ q|l for γ P p0, 1s, under these conditions ballistic behavior is fulfilled. More precisely, combining (0.1.9) and Theorem 3.3 of [Sz03] we have that all the conditions in Theorem 0.1.9 satisfy : P0 ´ a.s., Xn E0 rXτ1 | D “ 8s Ñv“ , n E0 rτ1 | D “ 8s 12 with v ¨ l ą 0. Furthermore the random variable τ1 has finite moments of any arbitrary order, and B¨n :“ Xr¨ ´ r¨ nsv ? nt ns converges in law on Skorohod space DpR` , Rd q under P0 to the law of a non-degenerate Brownian Motion with matrix covariance given by E0 rpXτ1 ´ τ1 vqt pXτ1 ´ τ1 vq | D “ 8s . A“ E0 rτ1 | D “ 8s It is then possible to show that the ballisticity conditions in direction l imply that P0 - a.s., Tul Ñ pv ¨ lq´1 as u Ñ 8. u Thus the walk escapes through direction l as if it had a local drift in direction l. Therefore the walk behavior is in concordance with the informal idea of what is meant by ballistic behavior. Besides seeing the effective criterion as a tool so as to get higher functional controls from lower ones on the walk exit probability by the unlikely side from slabs, we would like to mention that Sznitman in [Sz04] has found ballistic random walk exam1 ples satisfying pT q|l where the Kalikow’s condition breaks down. As a result Kalikow’s condition is not the weakest condition which ensures a ballistic behavior. Furthermore, 1 it is conjectured that pT q|l is equivalent to ballisticity in direction l, which implies that Conjecture 0.0.3 can be rephrased as: 1 pT q|l Ø the walk is transient in direction l. This ends our survey about ballisticity conditions in i.i.d. random environments. 0.2 Previous results for random walks in cone mixing random environments In this section we would like to mention some results for random walks in random environments which are not i.i.d. The main result of Chapter 2 of this thesis is formulated in a framework of random walks in random environments which satisfy a mixing condition 13 discussed in [CZ01], and called cone mixing condition. In [CZ01] it is proven that random walks in random environments satisfying a form of Kalikow’s condition, cone mixing, and some important additional assumptions, are ballistic. A similar result was obtained by Rassoul-Agha in [RA03], where he assumes also Kalikow’s condition and a mixing condition stronger that cone mixing called Dobrushin-Shlosman strong mixing assumption. Let us now describe these results. We first describe the main result in [CZ01], which ensures ballisticity under some conditions on the environment. Since mixing on cones is strictly weaker than the i.i.d. condition, it will not be surprising that we will have to strengthen the ballisticity conditions in order to ensure ballistic behavior. Even more, we will have to define approximate regeneration times, since the standard definition of them in the i.i.d. context does not work. For large fixed integer L we define τ1 pLq as the first time that the walk reaches a record level in direction l at time τ1 pLq ´ L, and such in the following L steps after this time, the walk does successive steps in the direction l. Further, after time τ1 pLq, the walk never exits the cone CpXn , l, αq again. This random time is much larger than the standard regeneration time used in the i.i.d. case. In fact, it can be shown that both τ1 pLq and Xτ1 pLq are of order κ´L as L Ñ 8. We also need to switch the stopping time D defined in (10) by D1 , which is essentially defined as the first exit time of the set Cp0, l, αq. We now need a suitable extension of the Kalikow’s condition. For V a finite, connected subset of Zd , with 0 P V , we let FV c “ σtωpz, ¨q : z R V u. The Kalikow’s random walk tXn : n ě 0u with state space V Y BV and starting from y P V Y BV is defined by the transition probabilities $ & řTV c E0 r n“0 1tXn “xu ωpx,eq|FV c s , řTV c E0 r n“0 1tXn “xu |FV c s for x P V and e P U % 1 for x P BV and e “ 0. PpV px, x ` eq :“ We denote by P̂y,V the law of this random walk and by Êy,V the corresponding expectation. The following extension of the Kalikow’s condition was introduced in [CZ01]. Definition 0.2.1 (Kalikow’s conditional condition). Let δ ą 0. We say that Kalikow’s 14 conditional condition with respect to the direction l is satisfied if there exists a positive constant δ such that inf V :xPV dpV pxq ¨ l ě δ, where px,V rX1 ´ X0 s “ dpV pxq :“ E ÿ ePpV px, x ` eq |e|“1 denotes the drift of Kalikow’s random walk at x, and the infimum runs over all finite connected subset V of Zd such that 0 P V . We denote this condition by pKCqδ . Finally, we set: F0,L " * L :“ σ ωpy, ¨q; y ¨ l ď ´ |u|2 . |u|1 The main result in [CZ01] is the following. Theorem 0.2.2 (Comets and Zeitouni). Consider a random walk in a random environment satisfying Kalikow’s conditional condition pKCqδ , the cone mixing condition pCM qα,φ |l and the ellipticity assumption pU Eq|l. Assume also that there exists a positive function M pLq depending just on L such that for some ϑ ą 1 one has that PrE0 rpκL τ1 qϑ | F0,L s ą M s “ 0 1 (13) 1 and satisfying limLÑ8 M pLq ϑ1 φ1 pLq α “ 0, where ϑ1 :“ ϑ{pϑ ´ 1q along with φ1 :“ P0 rD1 2φ . “ 8s ´ φ Then there exists a deterministic v P Rd ´ t0u such that P0 - a.s. lim Xn Ñ v, n with v ¨ l ą 0. The integrability condition (13) is essentially required in order to establish a law of large numbers along a regeneration time sequence which is not i.i.d. In the i.i.d. case, and 15 under the polynomial ballisticity condition pP ˚qM |l when M ě 15d ` 5 (c.f. Theorem), it is satisfied (as a matter of fact any moment of τ1 is finite). Nevertheless, the integrability condition (13) is quite unsatisfactory, since it is in general difficult to check wether a given random environments satisfies it or not. As a mater of fact, in [CZ01], a non-trivial example which satisfies (13) is given, but the argument presented there is not completely clear. On the other hand, Rassoul-Agha in [RA03], under a mixing condition called DobrushinShlosman strong mixing assumption (see [CZ01] or [RA03]) has proved ballistic behavior by means of a clever application of the environment as seen from the random walk technique. It is important to stand out that Rassoul-Agha has only assumed the usual Kalikow’s condition. However it was mentioned above that Kalikow’s condition is strictly stronger than condition pT 1 q [Sz04]. On the other hand, further important results can be found for instance in [CZ02], [RA05] and [G14]. In [CZ02] the authors proved suitable versions of the central limit theorem for the random walk in two kinds of environments: cone mixing and Dobrushing strong mixing. In [RA05], the author has investigated conditional versions of the strong law of large numbers. There it is proved that under an elliptic assumption and DobrushinShlosman strong mixing condition on the environment a weak version of the strong law of large number is satisfied. Finally, Guo in [G14] under similar assumptions gave an alternative proof of the result in [RA05] by means of regeneration arguments (instead of the theoretical tool used by Rassoul-Agha: the environment as seen from the random walk) and proved that there is at most one nonzero limit velocity when d ě 5 (originally proved in the i.i.d. case by Berger in [Be08]). In conclusion, in both of the articles [CZ01] and [RA03] some version of Kalikow’s condition is assumed. Furthermore, neither these works nor the ones mentioned in the last paragraph, discuss possible adaptations of weaker ballisticity conditions like conditions pT q, pT 1 q or pP qM , to environments which are not necessarily i.i.d., even less so asymptotic results under these kind of conditions. One of the objectives of this thesis, developed in Chapter 2, is to give a first indication about how should these ballisticity conditions be defined for cone mixing environments. 16 0.3 A brief explanation of the Thesis Results In this section we will describe the results of each chapter. 0.3.1 Main Result for i.i.d. Random Environments A problem left untouched in the quoted results of Section 0.1.2 is the following question: Conjecture 0.3.1. For a random walk in a uniformly elliptic i.i.d. environment, condi1 tion pT q|l is equivalent to pT q|l. Chapter 1 of this thesis addresses this question. The main result of Chapter 1 shows that condition pT 1 q|l implies an almost exponential decay for the exit probability of the random walk through the back side of slabs (which is very close to pT q|l). Specifically, for a given direction l and L ą 0 we denote by Sl,L the strip tx P Rd : |x ¨ l| ď Lu and by Al,L the event that the walk starting from 0 exits Sl,L through the side of Sl,L where x ¨ l ă 0. Now, for a given direction l and function γ : r0, 8q Ñ r0, 1s we say that the condition pT qγpLq |l is satisfied if for all directions l1 in a neighborhood of l there is a constant c ą 0 such that asymptotically as L Ñ 8 it is true that γpLq `opLγpLq q P0 rAl1 ,L s “ e´cL . It is straightforward to check that by definition, condition pT q|l is equivalent to pT qγ1 pLq |l with γ1 pLq “ 1 ´ C 1 , logpLq for any C ą 0. On the other hand, in [Sz03] Sznitman proved that pT 1 q|l implies pT qγ2 pLq |l with 1 log 2 pLq γ2 pLq “ 1 ´ C . logpLq In Chapter 1 of this thesis, we prove that pT 1 q|l implies pT qγ3 pLq |l with r γ3 pLq “ 1 ´ C 17 lognpLq pLq , logpLq (14) r is a positive constant and npLq a function with values in the positive integers where C that has limit infinity as L Ñ 8. logk denotes the function logarithm composed k ´ 1 k hkkkkkkkkkkkkkikkkkkkkkkkkkkj times with itself; i.e., logk pxq “ log ˝ log ˝ log ˝ . . . ˝ logpxq. In spite that this result seems to be close to answering affirmatively Conjecture 0.3.1, it does not. Indeed, the function npLq of (14) is such that lim lognpLq pLq “ 8. LÑ8 The proof of this result relies on renormalization arguments which have the Effective Criterion as a seed condition. 0.3.2 Main Result for cone mixing random environments Chapter 2 of this thesis is concerned with random walks in cone mixing random environments. The main result is the proof that under a non-effective polynomial ballisticity condition, these random walks have an asymptotic direction (see ??). In what follows we will define this version of the polynomial ballisticity ccondition. Given L, L1 ą 0, x P Zd , we define the boxes B L,L1 ,l ¯ ´ 1 1 d´1 X Zd , p0q :“ R p´L, Lq ˆ p´L , L q where R is a rotation on Rd such that Rpe1 q “ l. Define the positive boundary of BL,L1 ,l pxq, denoted by B ` BL,L1 ,l p0q, as B ` BL,L1 ,l p0q :“ BBL,L1 ,l p0q X tz : z ¨ l ě Lu, Define also the half-space Hx,l :“ ty P Zd : y ¨ l ă x ¨ lu, and the corresponding σ-algebra of the environment on that half-space Hx,l :“ σpωpyq : y P Hx,l q. Now, for M ě 1, we say that the non-effective polynomial condition pP CqM,c |l is satisfied if there exists some c ą 0 so that for y P H0,l one has that 18 ” ı lim LM sup P0 XTBL,cL,l p0q R B ` BL,cL,l p0q, TBL,cL,l p0q ă THy,l |Hy,l “ 0, LÑ8 (15) where the supremum is taken over all possible environments to the left of y¨l. We prove the existence of an asymptotic direction for random walks in random environment satisfying the condition pCM qα,φ |l under the assumptions pP CqM,c |l and pU Eq|l, where the positive constants M, c and α satisfy the constraints: 1 1 u M ą 6d and 0 ă α ď mint , 9 2c ` 1 (16) We will prove that the non-effective polynomial condition is weaker than the conditional version of Kalikow’s condition introduced in [CZ01]. We would like to sketch the general strategy behind the proof of this result. As a first step we need to prove that: P0 rD1 “ 8s ą 0. (17) Let us remark that we do not need a conditional version of the ballisticity assumption to prove this. To prove the claim (17), we have used renormalization type methods, so as to apply the polynomial condition. Specifically, using the assumption pU Eq|l we can and we do assume that the walk starting from 0 goes on a large distance through direction l up to a fixed point z with positive annealed probability, and starting from that point one can show that with a high probability the walk remains forever inside of each half-space: H˘i :“ ty : py ´ zq ¨ l˘i ě 0u, for i P r2, ds. Finally the result follows from the definition of the cone. We refer to the proof of Proposition 2.4.1 in Chapter 2 for the precise argument. As a second step, we proved a strong integrability result of the regeneration position Xτ1 pLq . Roughly speaking, we have proved that the conditional expectation of the second moment of the regeneration position is finite. These two steps are the core of the proof. Indeed using for instance similar arguments as the ones given in [CZ01] we can obtain the asymptotic direction vp. The main issue to integrate the second moment of the random variable κL Xτ1 pLq was to connect i.i.d. methods with the cone mixing model. We connect them by identifying how close (or far) the old τ1 is from the new τ1 pLq. The precise statement of the required integrability condition and its proof are given in Section 2.5 of Chapter 2. 19 On the other hand, the simpler Simenhaus’s approach [Si07] does not work in cone mixing environment at least if we identify the random variable τ1 with τ1 pLq. The argument of [Si07] makes a strong use of i.i.d. assumption on the environment. The i.i.d. structure of the environment space is explicitly required in the renewal theorem to prove Zerner’s formula (c.f. Lemma 2 in [Si07]) and in order to prove that the sequence pZk q :“ supně0 |Xn^τk`1 ´ Xτk | is such that Zn {n converges P0 -a.s. to 0 as n Ñ 8. The first argument breaks down in the cone-mixing case, mainly because one cannot apply the renewal theorem without assuming some kind of strong integrability condition for the regeneration position. Furthermore, as an example to understand possible pathologies in the behavior of a random walk in a cone mixing environment, we provided an example of a random walk defined in a cone mixing environment which is directionally transient but not ballistic, showing that we cannot expect the ballisticity conjecture 0.0.3 to be valid outside of the i.i.d. setting. Consequently one could ask the following: what would be the kind of natural conditions which ensure that the random walk satisfies a strong law of large numbers with a non-vanishing limit velocity in this framework? We expect that the machinery developed in Chapter 2 could serve in a future work to prove ballistic behavior under a ballisticity condition similar to condition pT 1 q. The two results are a joint work with Alejandro Ramı́rez. 20 Chapter 1 Almost exponential decay for the exit probability from slabs of ballistic RWRE 1.1 Introduction The relationship between directional transience and ballisticity for random walks in random environment is one of the most challenging open questions within the field of random media. In the case of random walks in an i.i.d. random environment, several ballisticity conditions have been introduced which quantify the exit probability of the random walk through a given side of a slab as its width L grows, with the objective of understanding the above relation. Examples of these ballisticity conditions include Sznitman’s pT 1 q and pT q conditions [Sz02, Sz03]: given a slab of width L orthogonal to l, condition pT 1 q in direction l is the requirement that the annealed exit probability of the walk through the γ side of the slab in the half-space tx : x ¨ l ă 0u, decays faster than e´CL for all γ P p0, 1q and some constant C ą 0, while condition pT q in direction l is the requirement that the decay is exponential e´CL . It is believed that condition pT 1 q, is equivalent to condition pT q. In this chapter we prove that condition pT 1 q implies an almost exponential decay (see Theorem 1.1.2 for the precise meaning of this statement) of the corresponding exit probabilities. Our proof relies on a recursive renormalization scheme, where the a careful 21 choice of fastly growing scales enables us to obtain the result. We use the equivalence between condition pT 1 q [Sz03] and the d ě 2 dimensional version of Solomon’s criterion [So75], known as the effective criterion [Sz03]. Let us introduce the random walk in random environment model. For x P Zd denote its euclidean norm by |x|2 . Let V :“ te P Zd : |e|2 “ 1u be the set of canonical vectors. Introduce the set P whose elements are 2d´vectors ppeqe P Zd , |e|“1 such that ppeq ě 0, for all e P V , ÿ ppeq “ 1. e P Zd , |e|“1 d We define an environment ω :“ tωpxq : x P Zd u as an element of Ω :“ P Z , where for each x P Zd , ωpxq “ tωpx, eq : e P V u P P. Consider a probability measure P on Ω endowed with its canonical product σ-algebra, so that an environment is now a random variable such that the coordinates ωpxq are i.i.d. under P. The random walk in the random environment ω starting from x P Zd is the canonical Markov Chain tXn : n ě 0u on pZd qN with quenched law Px,ω starting from x, defined by the transition probabilities for each e P Zd with |e| “ 1 by Px,ω rXn`1 “ Xn ` e|X0 , . . . , Xn s “ ωpXn , eq and Px,ω rX0 “ xs “ 1. The averaged or annealed law, Px , is defined as the semi-direct product measure Px “ P ˆ Px,ω on Ω ˆ pZd qN . Whenever there is a κ ą 0 such that inf ωpx, eq ě κ e,x P ´ a.s. we will say that the law P of the environment is uniformly elliptic. For the statement of the result, we need some further definitions. For each subset A Ă Zd we define the first exit time of the random walk from A as 22 TA :“ inftn ě 0 : Xn R Au. ´ :“ tx P Zd : x ¨ l ă uu, Fix a vector l P Sd´1 and u P R then define the half-spaces Hu,l ` Hu,l :“ tx P Zd : x ¨ l ą uu, Tul :“ TH ´ “ inftn ě 0, Xn ¨ l ě uu u,l and T̃ul :“ TH ` “ inftn ě 0, Xn ¨ l ď uu. u,l For γ P p0, 1s, we say that condition pT qγ |l holds with respect to direction l P Sd´1 , if 1 1 l lim sup L´γ log P0 rT̃´L ă TLl s ă 0, LÑ8 for all l1 in some neighborhood of l. Furthermore, we define pT 1 q|l as the requirement that condition pT qγ |l is satisfied for all γ P p0, 1q and condition pT q|l as the requirement that pT q1 |l is satisfied. In [Sz03], Sznitman proved that when d ě 2 for every γ P p0.5, 1q, pT qγ |l is equivalent to pT 1 q|l. This equivalence was improved in [DR11] and [DR12] culminating with the work of Berger, Drewitz and Ramı́rez who in [BDR14] showed that for any γ P p0, 1q, condition pT qγ |l implies pT 1 q|l. As a matter of fact, in [BDR14], an effective ballisticity condition, which requires polynomial decay was introduced. To define this condition, consider L, L̃ ą 0 and l P Sd´1 and the box Bl,L,L̃ ´ ¯ d´1 :“ R p´L, Lq ˆ p´L̃, L̃q X Zd , where R is a rotation defined by Rpe1 q “ l. (1.1) Given M ě 1 and L ě 2, we say that the polynomial condition pP qM in direction l (also denoted by pP qM |l) is satisfied on a box of size L if there exists and L̃ ď 70L3 such that ” P0 XTB l,L,L̃ ı 1 ¨l ă L ď M. L 23 Berger, Drewitz and Ramı́rez proved in [BDR14] that there exists a constant c0 such that whenever M ě 15d ` 5, the polynomial condition pP qM |l on a box of size L ě c0 is equivalent to condition pT 1 q|l (see also Lemma 3.1 of [CR14]). On the other hand, the following is still open. Conjecture 1.1.1. Consider a random walk in a uniformly elliptic random environment in dimension d ě 2 and l P Sd´1 . Then, condition pT q|l is equivalent to pT 1 q|l. To quantify how far are we presently from proving Conjecture 1.1.1, we will introduce now a family of intermediate conditions between conditions pT 1 q and pT q. Let γpLq : r0, 8q Ñ r0, 1s, with limLÑ8 γpLq “ 1. Let l P Sd . We say that condition pT qγpLq |l is satisfied if 1 1 l lim sup L´γpLq log P0 rT̃´L ă TLl s ă 0, (1.2) LÑ8 for l1 in a neighborhood of l. We will call γpLq the effective parameter of condition pT qγpLq . Note that condition pT q is actually equivalent to pT qγpLq with an effective parameter given by γpLq “ 1 ´ C , log L (1.3) for any constant C ě 0. In 2002 Sznitman [Sz03] was able to prove that pT 1 q implies pT qγpLq with effective parameter γpLq “ 1 ´ C a log L, log L (1.4) for some constant C ą 0. In this chapter, we are able to show that condition pT 1 q implies condition pT qγpLq with an effective parameter γpLq which is closer to the effective parameter for condition pT q given by (1.3). This is the first result since the introduction of condition pT 1 q by Sznitman in 2002, which would give an indication that Conjecture 1.1.1 is true. To state it, let us introduce some notations. Throughout, for each n ě 1, we will use the standard notation 24 n hkkkkkkikkkkkkj log ˝ ¨ ¨ ¨ ˝ log x, for the composition of the logarithm function n times with itself, for all x in its domain, where the n superscript means that the composition is performed n times. Theorem 1.1.2. Let d ě 2, l P S d´1 and M ě 15d ` 5. Assume that condition pP qM |l is satisfied on a box of size L ě c0 . Then there exists a constant C ą 0 and a function npLq : r0, 8q Ñ N satisfying limLÑ8 npLq “ 8, such that condition pT qγpLq |l , c.f. (1.2), is satisfied with an effective parameter γpLq given by npLq hkkkkkkikkkkkkj C γpLq “ 1 ´ log ˝ ¨ ¨ ¨ ˝ log L. log L (1.5) Remark 1.1.3. Note that the decay given by the effective parameter (1.5) of Theorem 1.1.2 is equivalent to the decay npLq´1 hkkkkkkikkkkkkj log ˝ ¨ ¨ ¨ ˝ log L 1 l1 lim sup log P0 rT̃´L ă TLl s ă 0, L LÑ8 for l1 in a neighborhood of l. Let us remark that a priori, even if npLq Ñ 8 as L Ñ 8, it might happen that the composition of the logarithm npLq time is bounded. Nevertheless, in the case of Theorem 1.1.2, it turns out that npLq hkkkkkkikkkkkkj lim log ˝ ¨ ¨ ¨ ˝ log L “ 8. LÑ8 Theorem 1.1.2 will be proven in the next section, but some remarks are in order. The strategy followed in the proof, roughly speaking, is to improve the iterative procedure used by Sznitman in [Sz02], to prove pT qγpLq with γpLq given by (1.4), through the so called effective criterion introduced by Sznitman in [Sz03]. The iterative procedure used in [Sz03], in spirit is a renormalization argument, where the idea is to control the exit probability of the walk recursively from an initial scale L0 to the final size of the slab 25 L ą L0 passing through a sequence of intermediate scales L0 ă L1 ă . . . ă Lk “ L. To go from scale L0 to scale L1 , a slab of width L1 is subdivided into overlapping slabs of width L0 , and the walk is looked at its exit times from successive slabs of width L0 . Essentially, at these times the walk looks like a one dimensional random walk in random environment, for which one can control its exit probabilities through the expected value of ρ, where ρ is close to the quotient between the probability to exit a slab of width L0 through its left side and the probability to exit it through its right side. Here, a triggering assumption is needed, which in our case is the effective criterion of Sznitman [Sz02] (the effective criterion is implied by the polynomial condition introduced by Berger, Drewitz and Ramı́rez in [BDR14]). This first step is the content of Proposition 2.1. A similar strategy is then used to pass from scale Lk to scale Lk`1 for k ě 1 (see Lemma 2.2). Nevertheless, reducing the movement of the random walk to a one dimensional walk, has a cost, which is a polynomial factor appearing in the recursion relations, and which somehow is the reason why one cannot go from the initial scale L0 directly to L in one step. In this chapter, we modify Sznitman’s argument, choosing a sequence of scales where Lk`1 is much larger than Lk compared to Sznitman’s approach, allowing us to work with a smaller number of intermediate steps in the recursion relation. The use of this new sequence of scales, produces at some points important difficulties in the proof which have to be properly handled. 1.2 Proof of Theorem 1.1.2 Throughout the rest of this section, we prove Theorem 1.1.2. Firstly, in subsection 1.2.1, we will introduce the basic notation which will be needed to implement the renormalization scheme, and we will recall a basic result of Sznitman which provides a bound for quantities involving the exit probability through the unlikely side of boxes which are inspired in techniques for used for one-dimensional random walks in random environment. In the second subsection, we will introduce a growth condition which will limit the maximal way in which the scales on the renormalization scheme can grow, while still giving a useful recurrence. In the third subsection we will choose an adequate sequence of scales satisfying the condition of subsection 1.2.2, and for which one can make computations. Finally, in 26 subsection 1.2.4, Theorem 1.1.2 will be proven using the scales constructed in subsection 1.2.3 through the use of the effective criterion [Sz02]. 1.2.1 Preliminaries and notation The proof of Theorem 1.1.2 will follow the renormalization method used by Sznitman to prove Proposition 2.3 of [Sz02]. The idea is to use a renormalization procedure which somehow mimics a computation for a one-dimensional random walk in random environment, where one goes from one scale to the next (larger) one through formulas where the exit probabilities of the random walk through slabs at the smaller scales are involved. Following Sznitman we introduce boxes transversal to direction l, which are specified in terms of B “ pR, L, L1 , L̃q, where L, L1 , L̃ are positive numbers and R is the rotation defined in (2.3). The box attached to B, is B :“ Rpp´L, L1 q ˆ p´L̃, L̃qd´1 q X Zd and the positive part of its boundary is defined as B` B :“ BB X tx P Zd , x ¨ l ě L1 , |Rpei q ¨ x| ă L̃, i ě 2u. We can now define the following random variable depending on a given specification B, analogous to the quotient in dimension d “ 1 between the probability to jump to the left and the probability to jump to the right [SW69, So75], for ω P Ω as ρB pωq :“ qB pωq , pB pωq where qB pωq :“ P0,ω rXTB R B` Bs “: 1 ´ pB pωq. The first step in the renormalization procedure will be to control the moments of ρB at the two first scales. To this end, consider positive numbers ? ? 3 d ă L0 ă L1 , 3 d ă L̃0 ă L̃1 along with the box-specifications 27 B0 :“ pR, L0 ´ 1, L0 ` 1, L̃0 q and B1 :“ pR, L1 ´ 1, L1 ` 1, L̃1 q. It is convenient to introduce now the notation q0 :“ qB0 , p0 :“ pB0 , q1 :“ qB1 , p1 :“ pB1 , and ρ0 :“ ρB0 , ρ1 :“ ρB1 . (1.6) Let also N0 :“ L1 L0 and Ñ0 :“ L̃1 . L̃0 We will also need to introduce the constant c1 pdq “ c1 :“ ? d. Note that for each pair of points x, y P Zd , there exists a nearest neighbor path joining them which has less than c1 |x ´ y|2 steps. Let us now recall the following Proposition of Sznitman [Sz03]. ? Proposition 1.2.1. There exist c2 pdq ą 3 d, c3 pdq, c4 pdq ą 1, such that when N0 ě 3, L0 ě c2 , L̃1 ě 48N0 L̃0 , for each a P p0, 1s one has that # ” a 2 ı ´10c1 L1 E ρ 1 ď c3 κ ` ř ¯ L̃1 ´ 3 12N0 L̃0 d´2 L1 c4 L̃1 L2 L̃0 Erq0 s 0 ´ ¯ rN0 s`m´1 2 d´1 a c4 L̃1 Erρ0 s 0ďmďN0 `1 28 + . (1.7) 1.2.2 The maximal growth condition on scales We next recursively iterate inequality (1.7) at different scales which will increase as fast as possible, in the sense that a certain induction condition should enable us to push forward the recursion. We next recursively iterate inequality (1.7) at different scales which will increase as fast as possible, in the sense that a certain induction hypothesis should enable us to push forward the recursion. Let v :“ 8, α :“ 240 and introduce two sequences of scales Lk , L̃k k ě 0, such that ? L0 ě c2 , 3 d ă L̃0 ď L30 (1.8) Nk ě 7, Lk`1 “ Nk Lk , L̃k`1 “ Nk3 L̃k , (1.9) and for k ě 0 as well as box-specifications Bk :“ pR, Lk ´ 1, Lk ` 1, L̃k q. Note that ˆ L̃k`1 “ Lk L0 ˙3 L̃0 . (1.10) Introduce also the notation for the respective attached random variables ρk :“ ρBk . Throughout, we will adopt the notation u0 :“ 3pd ´ 1q , L0 log κ1 and for k ě 1, 29 (1.11) uk :“ u0 . vk (1.12) We also let c5 :“ 2c3 c4 . Condition pGq. We say that the scales Lk , Nk , k ě 0 satisfy condition pGq if uk Nk ě αc1 for k ě 0, (1.13) and if 3pd´1q c5 Nk`1 uk`1 Lk`1 L3d´1 ď 1 for k ě 0. k`1 κ (1.14) Let us now state the following lemma which generalizes Lemma 2.2 of Sznitman ([Sz03]), for scales satisfying condition pGq. For completeness we include its proof. Lemma 1.2.2. Consider scales Lk , Nk , k ě 0, such that condition pGq is satisfied. Then, ? whenever L0 ě c2 , 3 d ď L̃0 ď L30 , and a0 P p0, 1s, we have that a0 u0 L0 ϕ0 :“ c4 L̃d´1 . 1 L0 Erρ0 s ď κ (1.15) d´1 ϕk :“ c4 L̃k`1 Lk Erρakk s ď κuk Lk . (1.16) then for all k ě 0, with ak “ a0 2´k , uk “ u0 v ´k . Proof. As in the proof of Lemma 2.2 of [Sz02], we can conclude by Proposition 1.2.1 that if L0 ě c2 (note that by the choice of Nk in (1.9), the other conditions of Proposition 1.2.1 are satisfied) we have that for k ě 0, 30 # + 2 Nk 12 rNk s`m´1 2 ÿ ´10c1 Lk`1 ϕk ` ϕk`1 ď c3 c4 L̃d´1 k`2 Lk`1 κ ϕk . (1.17) 0ďmďNk `1 We will now prove inequality (1.16) by induction on k using inequality (1.17). Since inequality (1.15) is identical to inequality (1.16) with k “ 0, the induction hypothesis is satisfied for k “ 0. We assume now that it is true for k ą 0, along with inequality (1.13) of assumption pGq and conclude that 2 Nk 2 Lk uk 24 κ´10c1 Lk`1 ϕk24 ď κ´10c1 Lk`1 κNk Nk 2 Therefore, using (1.18) and the fact that rNk s ´ 1 ě # ď 1. (1.18) because Nk ě 7 we see that + 2 Nk 24 Nk 4 ϕk`1 ď c3 c4 L̃d´1 k`2 Lk`1 ϕk ` Lk`1 ϕk Nk Nk 2 8 8 ď c5 L̃d´1 k`2 Lk`1 ϕk ϕk , (1.19) where we recall that c5 “ 2c3 c4 . Now, by the induction hypothesis (1.16) we see that Nk ϕk8 ď κuk`1 Lk`1 . Substituting this into (1.19), we see that it is enough now to show that Nk d´1 2 c5 L̃k`2 Lk`1 ϕk8 ď 1. But this is true, by (1.14) of condition pGq, the induction hypothesis and the inequality L̃k`1 ď L3k`1 for k ě 0 which follows by induction starting from (1.8). Indeed, using these facts, Nk 3pd´1q 2 8 ď c5 Nk`1 c5 L̃d´1 k`2 Lk`1 ϕk which ends the proof. 31 uk`1 Lk`1 L3d´1 ď 1, k`1 κ 1.2.3 An adequate choice of fast-growing scales We will now construct a sequence of scales tLk : k ě 0u which satisfy condition pGq, and for which Lemma 1.2.2 will eventually imply Theorem 1.1.2. This is not the fastest possible growing sequence of scales, but somehow it captures the best possible choice of γpLq. Let tfk : k ě 1u be a sequence of functions from r0, 8q to r0, 8q defined recursively as f0 pxq :“ 1, f1 pxq :“ v x and for k ě 1, fk`1 pxq :“ fk ˝ f1 pxq. Let now, for k ě 0, `“ k`1 ‰˘ f k`2 2 αc1 r 2 s `“ k ‰˘ . Nk :“ u0 fr k`1 s 2 (1.20) 2 According to display (1.9), we have the following formula valid for k ě 0, ˆ„ Lk`1 “ fr k`2 s 2 k`1 2 ˙ ˆ αc1 u0 ˙k`1 L0 . (1.21) Lemma 1.2.3. The condition uk Nk ě αc1 for k ě 0 (c.f. (1.13) of condition pGq) is equivalent to fr k`2 s `“ k`1 ‰˘ fr k`1 s `“ k ‰˘ 2 2 2 2 vk ě1 Furthermore, the last relation is fulfilled. 32 for k ě 0. (1.22) Proof. Note that (1.22) can be easily verified for k “ 0, 1 and 2. Therefore it is enough to prove inequality (1.22) for k ě 3. For this purpose, we will first show that for all positive integers n, and a, b P r1, 8q, we have that fn pa ` bq ě fn paqfn pbq. (1.23) To prove (1.23), suppose that A :“ tn P N : fn pa ` bq ă fn paqfn pbq for some a, b ě 1u ‰ ∅. Let m be the smallest element of A and remark that m is greater than 1. Also, note that fm pa ` bq ă fm paqfm pbq for some a, b ě 1. However, note that for a, b ě 1 one has that v a`b ě v a ` v b . Furthermore, for each k ě 0, the function fk p¨q is increasing. Therefore, fm´1 pv a qfm´1 pv b q “ fm paqfm pbq ą fm pa ` bq “ fm´1 pv a`b q ě fm´1 pv a ` v b q. This contradicts the minimality of m and hence A “ ∅ which proves (1.23). Back to (1.22), note that p1q f k`2 pr k`1 ´1 f f k`2 p1q pr k`1 2 sq r 2 s 2 s q r k`2 r 2 s r k`2 2 s 2 s ě f ě ě 1, k k v vk f k`1 pr k2 sqv k pr sq 2 r 2 s r k`1 2 s f where the first inequality was gotten using (1.23), the second one is a consequence of the inequality fr k`2 s `“ k`1 ‰ 2 fr k`1 s 2 ˘ ´ 1 2 `“ k ‰˘ ě 1, 2 33 valid for k ě 3, and which can be proved in a straightforward fashion if we divide the argument according to whether k is even or odd, and the last inequality comes from the fact that fr k`2 s´1 p1q ´ k ě 0 for k ě 3. 2 (1.24) Now, it is easy to verify inequality (1.24) when k “ 3 and k “ 4. Furthermore, the left hand of (1.24) is increasing as a function of k ě 2 for k odd. Similarly, it is increasing for k ě 2 for k even. We can therefore conclude, using induction that (1.24) is satisfied. This completes the proof of (1.22). Using Lemma 1.2.3 we can now obtain the following important lemma which gives conditions on the growth of a sequence of scale which ensure that pGq is satisfied. Lemma 1.2.4. There exists a constant c6 pdq such that when L0 ě c6 , the scales tLk : k ě 0u and tNk : k ě 0u defined by (1.21) and (1.20) satisfy condition pGq. Proof. By Lemma 1.2.3 we know that (1.13) of condition pGq is satisfied. We therefore just prove inequality (1.14) of condition pGq. We need to show that there exists a constant cpd, κq, such that whenever L0 ě cpd, κq, for all k ě 0 one has that 3pd´1q c5 Nk`1 3d´1 uk`1 Lk`1 Lk`1 κ ď 1. (1.25) We will first show that there exists c7 pd, κq “ c7 pdq ą 0, such that whenever L0 ě c7 , one has that for k ě 0, 3pd´1q Nk`1 κ uk`1 Lk`1 3 ď 1. (1.26) Now (1.26) is equivalent to ˜ 3pd ´ 1q logv pr k`2 2 sq r k`3 2 s f k`2 pr k`1 r s 2 sq f αc1 u0 ¸ 2 L0 u0 f ´ pr r k`2 2 s k`1 2 ´ sq αc1 vu0 3 34 ¯k`1 1 logv p κ q ď 0. Therefore, (1.26) is equivalent to the bound for k ě 0, ˜ ¸ k`2 k`3 pr 2 sq r s 2 1 logv αc u0 f k`2 pr k`1 sq r 2 s 2 L0 ě ´ `“ ‰˘ αc ¯k`1 `1˘. 1 fr k`2 s k`1 log v κ 2 vu0 f 9pd´1q u0 (1.27) 2 Let us focus on right-hand side of inequality (1.27) . Note that it can be split as ˜ f k`3 2 k`2 s pr 2 sq ¸ r ´ ¯ 9pd´1q logv f 1 u0 pr k`1 logv αc 2 sq u0 r k`2 2 s ` `1˘ `1˘. `“ k`1 ‰˘ ´ αc ¯k`1 `“ k`1 ‰˘ ´ αc ¯k`1 1 1 fr k`2 s logv κ fr k`2 s logv κ 2 vu0 2 vu0 2 2 9pd´1q u0 (1.28) Let us now try to find an upper bound for this expression independent on u0 (or equivalently, on L0 ). By the definition of u0 (c.f. (1.11)) note that for k ě 0 and L0 ě 3pd´1q 1 log κ one has that, 1 1 1 1 1 ´ ¯k`1 “ ´ ¯k ` αc1 ˘ ď ` αc ˘k`1 . 1 u0 αc1 αc1 v vu0 vu0 v Substituting this into (1.28) we see that it is bounded from above by ˜ ¸ f k`3 pr k`2 r 2 s 2 sq ´ ¯ 9pd ´ 1q logv f 1 pr k`1 9pd ´ 1q logv αc 2 sq u0 r k`2 2 s `“ k`1 ‰˘ ` αc ˘k`1 `1˘ ` `“ k`1 ‰˘ ` αc ˘k`1 `1˘. 1 1 fr k`2 s log f log k`2 v v 2 v κ 2 v κ r 2 s 2 (1.29) Note that only the left-most term of (1.29) depends on L0 . Choose a constant c8 pd, κq “ c8 pdq ą 1, such that if L0 ě c8 ˆ logv αc1 u0 ˙ ` ˘ logv κ1 ď L0 . d´1 (1.30) Then, when L0 ě c8 , we see using the fact that the left-most term of (1.29) is a decreasing function of k ě 0 and from inequality (1.30), that it can be bounded from above by L0 72 L0 9v ď L0 ď . αc1 240 3 35 (1.31) Thus, whenever L0 ě c8 , from (1.28), (1.29) and (1.31), we see that (1.27) is satisfied if ˜ ¸ f k`3 pr k`2 r 2 s 2 sq 9pd ´ 1q logv f pr k`1 2 sq 3 r k`2 2 s L0 ě (1.32) `“ k`1 ‰˘ ` αc ˘k`1 `1˘. 1 2 f k`2 log v 2 v κ r 2 s Therefore, in order to prove (1.26) it is enough to show that the right hand side of inequality (1.32) is bounded. To do this, it is enough to prove that the expression ˜ pr k`2 2 sq r k`3 2 s logv f pr k`1 2 sq r k`2 2 s `“ k`1 ‰˘ fr k`2 s 2 2 f ¸ is bounded. Now, ˜ pr k`2 2 sq r k`3 2 s logv f pr k`1 2 sq r k`2 2 s `“ k`1 ‰˘ fr k`2 s 2 2 f ¸ ´ `“ k`2 ‰˘¯ logv fr k`3 s 2 2 `“ k`1 ‰˘ ď . (1.33) fr k`2 s 2 2 “ k`3 ‰ “ k`2 ‰ “ ‰ “ k`2 ‰ Let us now remark that if k is even, then 2 “ 2 and k`1 “ 2 ´ 1. Therefore, 2 in this case, the right-hand side of inequality (1.33) is smaller than `“ ‰˘ `“ ‰˘ fr k`2 s´1 k`2 fr k`2 s´1 k`2 2 2 2 2 ´ k`2 ¯ . `“ k`2 ‰ ˘“ fr k`2 s ´1 fr k`2 s´1 v r 2 s´1 2 2 2 But, since for k fixed, the function fk p¨q is increasing, and since for k ě 0 we have that „ k`2 k ` 2 ´1 r s v 2 ě , 2 we see that the right-hand side of inequality (1.33) is bounded. Hence, for k even the right-most term of (1.33) is bounded by a constant c9 pd, κq “ c9 pdq ą 0. “ ‰ “ k`2 ‰ “ k`1 ‰ “ k`2 ‰ Suppose now that k is odd. Then k`3 “ ` 1 and “ 2 . Therefore, in 2 2 2 this case, the right-hand side of inequality (1.33) is equal to fr k`2 s 2 fr k`2 s 2 `“ k`2 ‰˘ 2 `“ k`2 ‰˘ “ 1, 2 so that we can conclude that the right-hand side of inequality (1.33) is bounded, and hence that there is constant c10 pd, κq “ c10 pdq ą 0 which is an upper bound for the right-hand 36 side of inequality (1.27). We can hence conclude, taking c7 pdq “ maxtc9 pdq, c10 pdqu, that when L0 ě c7 pdq, then (1.26) holds. As a second step to prove (1.25), we will show that it is possible to find a positive constant c11 pd, κq “ c11 pdq such that when L0 ě c11 one has that for all k ě 0, 3d´1 κ Lk`1 uk`1 Lk`1 3 ď 1. (1.34) Inserting the definition (1.21) that defines Lk into this inequality, we see that it is enough to prove that p3d ´ 1q logv pLk`1 q ´ If we show that for all k ě 0, L0 ě logv ´ `1˘ u0 κ αc1 u0 v ¯k`1 fr k`2 s `“ k`1 ‰˘ 2 2 L0 ď 0. 3 logv pLk`1 q3p3d´1q ¯ ´ , αc1 k`1 f k`2 pr k`1 u v 2 sq 1 logv p κ qu0 (1.35) we have a proof of (1.35). 0 r 2 s But the right-hand side of this inequality can be written as „ ´ ¯ k`1 ´ `“ ‰˘¯ 1 3p3d ´ 1q logv L0 αc 3p3d ´ 1q logv fr k`2 s k`1 u0 2 2 ‰˘ `“ k`1 . ` 1 ˘ ´ αc ¯k`1 `“ k`1 ‰˘ ` f k`2 1 2 r s logv κ u0 u0 v fr k`2 s 2 2 2 We need to establish a control with respect to L0 in this expression. Only the first term depends on L0 so we concentrate on the first term. Now, this term is decreasing with k. Therefore, it is smaller than ” ´ 3p3d ´ 1q logv L0 ` ˘` ˘ logv κ1 αcv 1 αc1 u0 ´ ¯ı “ 1 L20 αc1 logp κ q 3pd´1q ¯ 3p3d ´ 1q logv ` ˘` ˘ logv κ1 αcv 1 From this last expression, it is clear that we can choose a constant c12 pd, κq “ c12 pdq ą 0 such that whenever L0 ě c12 pdq one has that „ ´ ¯ k`1 1 3p3d ´ 1q logv L0 αc u0 L0 ď . ¯ ´ k`1 `1˘ `“ ‰˘ 3 αc1 k`1 logv κ u0 u0 v fr k`2 s 2 (1.36) 2 Therefore, if L0 ě c12 pdq and if ´ `“ k`1 ‰˘¯ 3p3d ´ 1q log f k`2 v 2 3 r 2 s `“ k`1 ‰˘ L0 ě , 2 fr k`2 s 2 2 37 (1.37) we would have (1.34), whenever we could prove that the right hand side of (1.37) is bounded independently of k ě 0. This can be proven in analogy to the previous computations made to show that the right-hand side of (1.32) is bounded. We have thus established the existence of a constant c11 pdq such that (1.34) is satisfied whenever L0 ě c11 pdq. On the other hand it is obvious that there is a constant c13 pdq, such that when L0 ě c13 pdq, for k ě 0, c5 κ uk`1 Lk`1 3 ď 1. Finally, in order for inequality (1.14) of condition pGq to be fulfilled, it is enough to take c6 pdq :“ maxtc7 pdq, c11 pdq, c13 pdqu. 1.2.4 The effective criterion implies Theorem 1.1.2 We continue now showing how Lemma 1.2.2 with the appropriate choice of scales, enables us to use the effective criterion (see Theorem 2.4 of [Sz03] where it was introduced) to prove the decay of Theorem 1.1.2. Let us define for x P Zd , |x|K :“ maxt|x ¨ Rpei q| : 2 ď i ď du. Also, define for each x P Zd , the canonical translation on the environments tx : Ω Ñ Ω as tx pωqpyq :“ ωpx ` yq for y P Zd . For the statement of the following proposition and its proof, we will use the shorthand notation for each n, pnq log8 pLq n hkkkkkkkikkkkkkkj :“ log8 ˝ ¨ ¨ ¨ ˝ log8 pLq. 38 ? Proposition 1.2.5. There exist c15 pdq ą 1, c14 pdq ě 3 d such that whenever L0 ě c14 , ? 3 d ď L̃0 ď L30 , and for the box specification B0 “ pR, L0 ´ 1, L0 ` 1, L̃0 q, the condition ˆ c15 ˆ ˙˙3pd´1q 1 3d´2 log L̃d´1 inf Erρa0 s ă 1, 0 L0 aPp0,1s κ (1.38) is satisfied (recall the definition of ρ0 in (1.6)), then there exist a constant c ą 0 and a function npLq : r0, 8q Ñ N, with npLq Ñ 8 as L Ñ 8, such that npLq lim sup L´1 exptc log8 LÑ8 l Lu log P0 pTLl ď T̃´L q ă 0. (1.39) Remark 1.2.6. The assumption (1.38) of Proposition 1.2.5, is called the effective criterion, and was introduced by Sznitman in [Sz03]. Proof. Let us choose a sequence of scales tLk : k ě 0u and tL̃k : k ě 0u according to displays (1.21) and (1.10). With this choice of scales, as in the proof of Proposition 2.3 of Sznitman [Sz03], one can see that there are constants c15 pdq and c14 ě maxtc6 , c2 u such that if L0 ě c14 then condition (1.38) implies condition (1.15) of Lemma 1.2.2 with u0 chosen according to (1.11). By Lemma (1.2.4), the chosen scales tLk : k ě 0u and tL̃k : k ě 0u satisfy condition pGq. Therefore, since (1.15) of Lemma (1.2.2) is satisfied , we know that for all k ě 0, inequality (1.16) is satisfied. The strategy to prove (1.39) will be similar to that employed in [Sz03] to prove Proposition 2.3: we will first choose an appropriate k so that Lk approximates a fixed scale L tending to 8. Nevertheless, since here we are working with scales which are much larger than those used in [Sz03], we will have to be much more careful with this argument. Let L ě L0 . Then, there exists a unique integer k “ kpLq such that Lk ď L ă Lk`1 . Note that to prove (1.39) it is enough to show that there exists a positive constant c16 such that for all L ě L0 one has that " " ** 1 1 pr k`1 2 sq ă ď exp ´c16 L exp ´ log8 pLq . c16 c16 “ ‰ In effect, since clearly k Ñ 8 as L Ñ 8, choosing npLq “ k`1 we have (1.39). 2 l P0 rT̃´L TLl s We will divide the proof of (1.40) into two cases. 39 (1.40) Case 1. Assume that Lď 2αc1 k v Lk . u0 (1.41) Let " „ * L d B :“ x P Z : |x|K ď L̃k , x ¨ l P p´L, Lq . Lk From the inequality Erqk s ď Erρakk s, Lemma 1.2.2 and Chebyshev inequality, we see that if 1 H :“ tω P Ω : Dx P B such that qk ˝ tx pωq ě κ 2 uk Lk u, then 1 PrHs ď κ 2 uk Lk |B| L̃d´1 k`1 Lk . Note that on Hc , by the strong Markov property one has that ” P0,ω rTLl ď l T̃´L s ě p1 ´ κ 1 u L 2 k k q L Lk ı `1 . Therefore, since for x P r0, 1s and n natural one has that p1 ´ xqn ď np1 ´ xq, for L large enough ˆ l P0 rT̃´L ă TLl s ď |B| L̃d´1 k`1 Lk ` L Lk ˙ 1 ` 1 κ 2 uk Lk ´ ¯d 1 ď 3 ˆ 2d LLk κ 2 uk Lk ¯ 1 ´ k d ď 3 ˆ 2d 2αcu10v κ 4 uk Lk ď 1, (1.42) where in the third inequality we have used our assumption on L (1.41). Hence, we can check that there is a constant c17 , such that for k ě 0, l P0 pT̃´L ă TLl q " * 1 Lk ď exp ´c17 k . c17 v (1.43) Now, again by our assumption (1.41), observe that there is a constant c18 such that 40 Lk L ą c18 2k . (1.44) k v b v On the other hand, note that when L0 ě αc3pd´1q 1 , we have by the choice of scales given 1 log κ in (1.21), that for k ě 1 ˆ„ ˙ k fr k`1 s ď Lk ď L. 2 2 Repeatedly taking logarithms in (1.45), we conclude that for k ě 1 „ k k pr k`1 sq ď ď log8 2 pLq. 4 2 (1.45) (1.46) Then, substituting the inequalities (1.44) and (1.46) into (1.43), we see that there exists a positive constants c16 such that for L ě L0 " " ** 1 1 pr k`1 2 sq ă ď exp ´c16 L exp ´ log8 pLq . c16 c16 “ ‰ Now, (1.39) follows taking npLq “ k`1 . 2 l P0 rT̃´L TLl s Case 2. Let us now assume that Lą 2αc1 k v Lk . u0 Let mk be the unique integer such that mk Lk ď L ă pmk ` 1qLk . By the definition of mk we have the inequality mk ě αc1 k v . u0 (1.47) We will now follow an approach similar to the one employed for Case 1, but using a sequence of scales which approximate L with a higher precision than the tLk u sequence. Let us define S1k :“ mk Lk , rk , Sr1k :“ m3k L S2k :“ m2k Lk , rk , Sr2k :“ m6k L 41 (1.48) along with the box-specification Bp :“ pR, S1k ´ 1, S1k ` 1, Sr1k q and the random variable ρpk attached to this box-specification. In analogy with the proof of Lemma 1.2.2, we will prove that a k pSr2k qd´1 S1k Erp ρkk`1 s ď κuk`1 S1 . (1.49) For the time being, assume that this inequality is true. Let " „ * L d k p B “ x P Z : |x|K ď S̃ , x ¨ l P p´L, Lq . S1k 1 In analogy with the development of Case 1, using (1.49) we can arrive to the following inequality analogous to (1.42) ˜ l P0 rT̃´L ă TLl s ď ¸ 1 L k ` k ` 1 κ 2 uk`1 S1 . k d´1 k r S1 pS2 q S1 p |B| From here we conclude that there is a constant c19 such that for k ě 0 * " c19 S1k 1 l l exp ´ k P0 rT̃´L ă TL s ď c19 v (1.50) u0 Now, the computation S1k “ mk Lk “ pmk ` 1qLk ´ Lk ě L ´ 2αc v ´k L, replaced at (1.50), 1 gives us l P0 rT̃´L ´ ¯, $ u0 ´k . & c L 1 ´ v 19 2αc1 1 ă TLl s ď exp ´ % c19 vk So that, there exists c20 such that l P0 pT̃´L ă TLl q * " 1 L ď exp ´c20 k c20 v Using now (1.46) we conclude that there is a constant c16 such that for L ě L0 one has that l P0 rT̃´L Choosing npLq “ ă “ k`1 ‰ 2 TLl s " " ** 1 1 pr k`1 2 sq exp ´c16 L exp ´ log8 ď pLq . c16 c16 we conclude the proof. Now, we need to prove (1.49). Using Proposition 1.2.1, with Bp and Bk instead of B1 and B0 , we have: 42 # ´10c1 S1k a ρkk`1 s ď c3 κ Erp m2 k 12 ϕk + mk `j´1 2 ÿ ϕk ` 0ďjďmk `1 So that # a ρkk`1 s pSr2k qd´1 S1k Erp ď c3 pS2k qd´1 S1k ´10c1 S1k κ m2 k 12 ϕk + mk `j´1 2 ÿ ϕk ` 0ďjďmk `1 Now, ´10c1 S1k κ m2 k 24 ϕk k ď κ´10c1 S1 κ ku mk S1 k 24 ď 1. (1.51) where the first inequality follows from inequality (1.47), the definition (1.48) of S1k and (1.12) of uk , and from Lemma 1.2.4, which enables us to apply inequality (1.16) of Lemma 1.2.2, while the second inequality of (1.51) follows from the fact that mk uk ě 240c1 for k ě 0. Then, inequality (1.51) and the fact that mk ´ 1 ě # a pSr2k qd´1 S1k Erp ρkk`1 s ď c3 pSr2k qd´1 S1k mk , 2 m2 k 24 ϕk imply that + mk 4 ` S1k ϕk . So that mk a k pSr2k qd´1 S1k Erp ρkk`1 s ď 2c3 pSr2k qd´1 pS1k q2 ϕk 8 κuk`1 S1 . Where, it was used the result of Lemma 1.2.2. Finally, note that to finish the proof we have to show that for k ě 0, mk 2c3 pS̃2k qd´1 pS1k q2 ϕk 8 ď 1. (1.52) By our definitions in (1.48), rd´1 L2 . pSr2k qd´1 pS1k q2 “ m6d´4 L k k k mk ` ˘ mk Now, by Lemma 1.2.4 and its consequence Lemma 1.2.2, we have that ϕk 8 ď κuk Lk 8 “ κuk`1 mk Lk . Therefore, the left hand side of inequality (1.52) is smaller than 2 uk`1 mk Lk 2ck mk6d´4 L̃d´1 . k Lk κ 43 However, as d is fixed, and k is large, it is clear that 2 L̃d´1 k Lk κ uk`1 mk Lk 2 ď1 and 2c3 mk6d´4 κ uk`1 mk Lk 2 ď 1. This completes the proof. It is now easy to check that Proposition 1.2.5 implies Theorem 1.1.2 with the function log x replaced by log8 x. Indeed, note that (1.38) is equivalent to the effective criterion. On the other hand, using the fact that for every x ą 0, log x ě log8 x, we can then obtain Theorem 1.1.2. Remark 1.2.7. Let us remark that somehow our choice of scales is optimal. More precisely, in this chapter we have tacitly assumed that the estimate in Proposition 1.2.1 cannot be improved in asymptotic terms. Once this is assumed, the requirements: (1.13) and (1.14) are sharp inequalities and one can verify that the inequality (1.33) is not satisfied for the choice of scales determined by `“ k`2 ‰˘ f k`2 2 αc1 r 2 s `“ k`1 ‰˘ , Nk1 “ u0 fr k`1 s 2 2 which implies: the scales tNk1 ukě0 do not satisfy condition pGq. On the other hand, it is clear that the previous scale would give us the same result as in Theorem 1.1.2. Therefore a new idea should be introduced to prove the Conjecture 1.1.1. Acknowledgments: We thank A.-S. Sznitman for suggesting to explore how close can one get to the exponential decay of condition pT 1 q from the effective criterion via renormalization. 44 Chapter 2 Asymptotic Direction for Random Walk in Strong Mixing Environment 2.1 Introduction Random walk in random environment is basic model of statistical mechanics while challenging questions about it remain open (see [Ze1] for a general overview). It is a simple but powerful model for a variety of phenomena including homogenization in disordered materials [M94], DNA chain replication [Ch62], crystal growth [T69] and turbulent behavior in fluids [?]. In the multidimensional setting a a widely open question is to establish relations between the environment at a local level and the long time behavior of the random walk. During last ten years, interesting progress has been achieved specially in the case in which the movement takes place on the hyper-cubic lattice Zd and the environment is i.i.d., establishing relations between directional transience, ballisticity and the existence of an asymptotic direction and the law of the environment in finite regions. Nevertheless, to a great extent, these arguments are no longer valid when the i.i.d. assumption is dropped. In this chapter we focus on the problem of finding local conditions on the environment which ensure the existence of an asymptotic direction for the random walk in contexts where the environment satisfies some mixing condition, but it is not necessarily i.i.d. To be more precise, we establish the existence of an asymptotic direction for random walks in random environments which are uniformly elliptic, are cone mixing [CZ01], and satisfy 45 a non-effective version of the polynomial ballisticity condition introduced in [BDR14]. While they are directionally transient, these random walks may have a vanishing velocity even for dimensions d ą 1. For x P Rd , we denote by |x|1 , |x|2 and |x|8 its l1 , l2 and l8 norms respectively. For each integer d ě 1, we consider the p2d ´ 1q-dimensional simplex Pd :“ tz P pR` q2d : ř2d d Zd i“1 zi “ 1u and E :“ te P Z : |e|1 “ 1u. We define the environmental space Ω :“ Pd and endow it with its product σ-algebra. Now, for a fixed ω “ tωpyq : y P Zd u P Ω, with ωpyq “ tωpy, eq : e P U u P Pd , and a fixed x P Zd , we consider the Markov chain tXn : n ě 0u with state space Zd starting from x defined by the transition probabilities Px,ω rXn`1 “ Xn ` e | Xn s “ ωpXn , eq for e P U. (2.1) We denote by Px,ω the law of this Markov chain and call it a random walk in the environment ω. Consider a law P defined on Ω. We call Px,ω the quenched law of the random walk starting from x. Furthermore, we define the semi-direct product probability measure on Ω ˆ pZd qN by ż Px,ω pBqdP Px pA ˆ Bq :“ A for each Borel-measurable set A in Ω and B in pZd qN , and call it the annealed or averaged law of the random walk in random environment. The law P of the environment is said to be i.i.d. if the random variables tωpxq : x P Zd u are i.i.d. under P, elliptic if for every x P Zd and e P U one has that Prωpx, eq ą 0s “ 1 while uniformly elliptic if there exists a κ ą 0 such that Prωpx, eq ě κs “ 1 for every x P Zd and e P U . Let l P Sd´1 . We say that a random walk is transient in direction l or just directionally transient if P0 -a.s. one has that lim Xn ¨ l “ 8. nÑ8 Furthermore, we say that it is ballistic in direction l lim inf nÑ8 Xn ¨ l ą 0. n 46 In the case in which the environment is elliptic and i.i.d., it is known that whenever a random walk is ballistic necessarily a law of large numbers is satisfied and in fact limnÑ8 Xn n “ v ‰ 0 is deterministic [DR14]. Furthermore, in the uniformly elliptic i.i.d. case, it is still an open question to establish wether or not in dimensions d ě 2, every directionally transient random walk is ballistic (see [BDR14]). On the other hand, we say that v̂ P Sd´1 is an asymptotic direction if P0 -a.s. one has that Xn “ v̂. nÑ8 |Xn |2 lim For elliptic i.i.d. environments, Simenhaus established [Si07] the existence of an asymptotic direction whenever the random walk is directionally transient in an open set of Sd´1 . As it will be shown in Section 2.3, this statement is not true anymore when the environment is assumed to be ergodic instead of i.i.d., even if it is uniformly elliptic. In this chapter we establish the existence of an asymptotic direction under three assumptions about the law P of the environment: a weak form of uniform ellipticity; cone mixing; a ballisticity condition demanding polynomial decay with high enough degree of the annealed exit probability of the random walk from the back and lateral side of boxes. All these assumption will be defined with respect to a fixed direction l P Sd´1 . It will be shown in section 2.3, that there exist environments almost satisfying the above assumptions which are directionally transient but not ballistic. Here the term almost is used because in these examples the polynomial ballisticity condition is satisfied with a low degree. Let us describe these assumption with more precision. Let κ ą 0. We say that P is uniformly elliptic with respect to l, denoted by pU Eq|l, if the jump probabilities of the random walk are positive and larger than 2κ in those directions which for which the projection on l is positive. In other words if Prωp0, eq ą 0s “ 1 for e P E and if ” ı P min ωp0, eq ě 2κ “ 1, ePE where 47 E :“ Ydi“1 tsgnpli qei u ´ t0u (2.2) and by convention sgnp0q “ 0. We will now introduce a certain mixing assumption for the environment P. Let α ą 0 and R be a rotation such that Rpe1 q “ l. (2.3) To define the cone, it will be useful to consider for each i P r2, ds, l`i “ l ` αRpei q |l ` αRpei q| and l´i “ l ´ αRpei q . |l ´ αRpei q| The cone Cpx, l, αq centered in x P Rd is defined as Cpx, l, αqq :“ d č ( z P Rd : pz ´ xq ¨ l`i ě 0, pz ´ xq ¨ l´i ě 0 . (2.4) i“2 Let φ : r0, 8q Ñ r0, 8q be such that limrÑ8 φprq “ 0. We say that a stationary probability measure P satisfies the cone mixing assumption with respect to α, l and φ, denoted pCM qα,φ |l, if for every pair of events A, B, where PpAq ą 0, A P σtωpz, ¨q; z ¨ l ď 0u, and B P σtωpz, ¨q; z P Cprl, l, αqu, it holds that ˇ ˇ ˇ PrA X Bs ˇ ˇ ˇ ď φpr|l|1 q. ´ PrBs ˇ PrAs ˇ We will see that every stationary cone mixing measure P is necessarily ergodic. On the other hand, a cone-mixing environment can be such that the jump probabilities are highly dependent along certain directions. We now introduce an assumption which is closely related to the effective polynomial ballistic condition introduced in [BDR14]. For each A Ă Zd we define BA :“ tz P Zd : z R A, there exists some y P A such that |y ´ z| “ 1u. Define also the stopping time TA :“ inftn ě 0 : Xn R Au. 48 Given L, L1 ą 0, x P Zd and l P Sd´1 we define the boxes ´ ¯ 1 1 d´1 BL,L1 ,l pxq :“ x ` R p´L, Lq ˆ p´L , L q X Zd , where R is defined in (2.3). The positive boundary of BL,L1 ,l pxq, denoted by B ` BL,L1 ,l p0q, is B ` BL,L1 ,l p0q :“ BBL,L1 ,l p0q X tz : z ¨ l ě Lu, Define also the half-space Hx,l :“ ty P Zd : y ¨ l ă x ¨ lu, and the corresponding σ-algebra of the environment on that half-space Hx,l :“ σpωpyq : y P Hx,l q. Now, for M ě 1, we say that the non-effective polynomial condition pP CqM,c |l is satisfied if there exists some c ą 0 so that for y P H0,l one has that ” ı lim LM sup P0 XTBL,cL,l p0q R B ` BL,cL,l p0q, TBL,cL,l p0q ă THy,l |Hy,l “ 0, LÑ8 (2.5) where the supremum is taken over all possible environments to the left of y ¨ l. It can be verified that for i.i.d. environments, this condition is implied by Sznitman’s pT 1 q condition [Sz03], and it is implied by the effective polynomial condition introduced in [BDR14]. Throughout this chapter, we will denote by S˚ d´1 the subset of Sd´1 defined by S˚ d´1 :“ ts P Sd´1 : there exists y P R ´ t0u, such that ys P Zd u. We can now state our main result. Theorem 2.1.1. Let l P S˚ d´1 X, M ą 6d, c ą 0 and 0 ă α ď mint 91 , 1 u. 2c`1 Con- sider a random walk in a random environment with stationary law satisfying the uniform ellipticity condition pU Eq|l, the cone mixing condition pCM qα,φ |l and the non-effective polynomial condition pP CqM,c |l. Then, there exists a deterministic v̂ P Sd´1 such that P0 -a.s. one has that 49 Xn “ v̂. nÑ8 |Xn | lim As it will be explained in Section 2.3, Simenhaus’s theorem which states that an asymptotic direction exists whenever the random walk is directionally transient in an open set of directions and the environment is i.i.d., is not true if the i.i.d. assumption is dropped. Somehow, Theorem 2.1.1 shows that if the i.i.d. assumption is weakened to cone mixing, while directional transience is strengthened to the non-effective polynomial condition, we still can guarantee the existence of an asymptotic direction. In [CZ01], the existence of a strong law of large numbers is established for random walks in cone-mixing environments which also satisfy a version of Kalikow’s condition, but under an additional assumption of existence of certain moments of approximate regeneration times. This assumption is unsatisfactory in the sense that it is in general difficult to verify if for a given random environment it is true or not. On the other hand, as it will be shown in Section 2.3, there exist examples of random walks in a random environment satisfying the cone-mixing assumption for which the law of large numbers is not satisfied, while an asymptotic direction exists. From this point of view, Theorem 1.1 is also a first step in the direction of obtaining scaling limit theorems for random walks in cone-mixing environments through ballisticity conditions weaker than Kalikow’s condition, and without any kind of assumption on the moments of approximate regeneration times or of the position of the random walk at these times. On the other hand, in [RA03], a strong law of large numbers is proved for random walks which satisfy Kalikow’s condition and Dobrushin-Shlosman’s strong mixing assumption. The Dobrushin-Shlosman strong mixing assumption is stronger than cone-mixing, both because it implies cone-mixing in every direction and because it corresponds to a decay of correlations which is exponential. A key step to prove Theorem 1.1 will be to establish that the probability that the random walk never exits a cone is positive through the use of renormalization type ideas, and only assuming the non-effective polynomial condition and uniform ellipticity. Using this fact, we will define approximate regeneration times as in [CZ01], showing that they have finite moments of order larger than one when we also assume cone-mixing. This part of the proof will require careful and tedious computations. Once this is done, the existence 50 of an asymptotic direction can be deduced using for example the coupling approach of [CZ01]. We will now describe the general structure of the sections in this chapter. In Section 2.3, we will present two examples of random walks in random environments which exhibit a behavior which is not observed in the i.i.d. case, giving an idea of the kind of limitations given by the framework of Theorem 2.1.1. In Section 2.2, the meaning of the non-effective polynomial condition and its relation to other ballisticity conditions will be discussed. In Section 2.3, we will present two examples of random walks in random environments which exhibit a behavior which is not observed in the i.i.d. case, giving an idea of the kind of limitations given by the framework of Theorem 2.1.1. In Section 2.4, we will show that the non-effective polynomial condition implies that the probability that the random walk never exits a cone is positive. This will be used in Section 2.5 to prove that the approximate regeneration times have finite moments of order larger than one. Finally in Section 2.6, Theorem 2.1.1 will be proved using coupling with i.i.d. random variables. 2.2 2.2.1 Preliminary discussion Non-effective polynomial condition and its relation with other directional transience conditions Here we will discuss the relationship between the condition non-effective polynomial condition and other transience conditions. Furthermore we will show that the conditional non-effective polynomial condition is weaker than the conditional version of Kalikow’s condition introduced by Comets and Zeitouni in [CZ01]. For reasons that will become clear in the next section, the following definition, which is actually weaker than the conditional non-effective polynomial condition, will be useful. Let l P Sd´1 , M ě 1 and c ą 0. We say that condition pP qM,c |l is satisfied, and we call it the non-effective polynomial condition if there is a constant c ą 0 such that limLÑ8 LM P0 rXTBL,cL,l p0q R B ` BL,cL,l p0qs “ 0. 51 It is straightforward to see that pP CqM,c |l implies pP qM,c |l. It should be pointed out, that for a fixed γ P p0, 1q, if both in the conditional and non-conditional non-effective polynomial conditions the polynomial decay is replaced by γ a stronger stretched exponential decay of the form e´L , one would obtain a condition defined on rectangles equivalent to condition pT qγ introduced by Sznitman in [Sz03], and also a conditional version of it. On the other hand, as we will see now, the conditional non-effective polynomial condition is implied by Kalikow’s condition as defined in [CZ01] for environments which are not necessarily i.i.d. Let us recall this definition. For V a finite, connected subset of Zd , with 0 P V , we let FV c “ σtωpz, ¨q : z R V u. The Kalikow’s random walk tXn : n ě 0u with state space in V Y BV , starting from y P V Y BV is defined by the transition probabilities $ & řTV c E0 r n“0 1tXn “xu ωpx,eq|FV c s , řTV c E0 r n“0 1tXn “xu |FV c s for x P V and e P E % 1 for x P BV and e “ 0. PpV px, x ` eq :“ We denote by P̂y,V the law of this random walk and by Êy,V the corresponding expectation. The importance of Kalikow’s random walk stems from the fact that XTV c has the same law under Pp0,V and under P0 r¨|FV c s (2.6) (see ([K81])). Let l P Sd´1 . We now define Kalikow’s condition with respect to the direction l as the following requirement: there exits a positive constant δ such that inf V :xPV dpV pxq ¨ l ě δ, where px,V rX1 ´ X0 s “ dpV pxq :“ E ÿ ePpV px, x ` eq ePE denotes the drift of Kalikow’s random walk at x, and the infimum runs over all finite connected subset V of Zd such that 0 P V . The following result shows that Kalikow’s condition is indeed stronger that the conditional non-effective polynomial criteria. 52 Proposition 2.2.1. Let l P Sd´1 . Assume Kalikow’s condition with respect to l. Then there exists an r ą 0 such that for all y P H0,l one has that limLÑ8 L´1 sup log P0 rXTBL,rL,l p0q R B ` BL,rL,l p0q, TBL,rL,l p0q ă THy,l |Hy,l s ă 0, where the supremum is taken in the same sense as in (2.5). In particular, Kalikow’s condition with respect to direction l implies pP CqM,r |l for all M ą 0. Proof. Suppose that Kalikow’s condition is satisfied with constant δ ą 0. We will first assume that y ¨ l P p´L, 0q. Let c ą 1. For y P H0,l and L ě 1 consider the box ˆ ´ c c ¯d´1 ˙ V :“ R ry ¨ l, Ls ˆ ´ L, L . δ δ Therefore, using (2.6) we find that P0 rXTB L, c L,l δ p0q R B ` BL,cL,l p0q, TBL, c L,l p0q ă THy,l |FV c s ď P0 rXTUr ¨ Rpej q ě δ c L δ for some j P r2, ds, |XTUr ¨ l| ă L|FV c s “ Pp0,V rXTUr ¨ Rpej q ě δc L for some j P r2, ds, |XTUr ¨ l| ă Ls. (2.7) Notice that on the set c tXTV ¨ Rpej q ě L for some j, XTV ¨ l ă Lu, δ one has Pp0,V -a.s. that „ cL TV ě . δ Thus, by means of the auxiliary martingale tMnV : n ě 0u defined by MnV :“ Xn ´ X0 ´ n´1 ÿ dpV pXj q, j“0 which has bounded increments (indeed bounded by 2) we can see that on tTV ą “ cL ( s , δ we have that for L large enough that ˆ MrVcL s δ ¨l ăL´ ˙ cL p1 ´ cqL ´ 1 δ “ p1 ´ cqL ` δ ă δ 2 (2.8) Pp0,V -a.s. Now, it will be convenient at this point to recall Azuma’s inequality [Sz01]: 53 Pp0,V rMnV " * A2 ¨ w ą As ď exp ´ for A ą 0, n ě 0, |w| “ 1, 8n for martingales with increments bounded by 2. Using this inequality and (2.8) we obtain that Pp0,V rXTUr ¨ Rpej q ą δc L for some j, XTV ¨ l ď Ls ď Pp0,V rTV ą cL s δ ď Pp0,V rMrVcL s ¨ p´lq ą pc ´ 1qL{2s ď expt´c1 Lu, (2.9) δ for a suitable positive constant c1 . Finally, coming back to (2.7), we can then conclude that limLÑ8 L´1 sup log P0 rXTBL,rL,l p0q R B ` BL,rL,l p0q, TBL,rL,l p0q ă THy,l |Hy,l s ă 0, where r “ δc . Let us now assume that y ¨ l ď ´L. By Lemma 1.1 in [Sz01] we know that there exists a positive constant ψ depending on δ such that for all V finite connected subsets of Zd with 0 P V e´ψXn ¨l is a supermartingale with respect to the canonical filtration of the walk under Kalikow’s law Pp0,V . Thus, we have that Pp0,V rXTV ¨ l ď ´Ls ď expt´ψLu by means of stopping time theorem applied at time TV . By an argument similar to the one developed for the case y ¨ l P p´L, 0q, we can finish the estimate in the case y ¨ l ď L. 2.2.2 Cone mixing and ergodicity The main objective in this section is to establish the following: any stationary probability measure P defined on the canonical σ´ algebra F, which satisfies property pCM qφ,α |l is 54 ergodic with respect to space-shifts. Before doing this, let us recall an ergodic notion. We say that E P F is an invariant set if : θx E :“ E for all x P Zd . Theorem 2.2.2. Assume that the probability space pΩ, F, Pq has the property pCM qφ,α |l and is stationary, then the probability measure P is ergodic, i.e. for any invariant set E P F we have: PrEs P t0, 1u. Proof. Let E P F be an invariant set. From a theoretical measure fact, given ą 0 there exists a cylinder measurable set A P F, so that: PrA4Es ă . Since A is a cylinder measurable set, it can be represented as: A “ tωpx, ¨q : x P F, F Ă Zd , |F | ă 8, ωpxi , ¨q P Pi , for xi P F , Pi P BpPd qu, where as a matter of definition BpPd q stands for the borelian σ´ algebra on the compact subset Pd of R2d . We choose L such that: φpLq ă . Plainly, for L we can find an x P Zd such that θx A and A are L separated on cones with respect to direction l , in other words: There exists y P Zd such that: A P σtωpz, ¨q : z ¨ l ď y ¨ l ´ Lu along with θx A P σtωpz, ¨q : z P Cpy, l, αqu. 55 We can suppose that PrEs ą 0, otherwise there is nothing to prove. So as to complete the proof we have to show that PrEs “ 1. Therefore taking small enough we can suppose further PrAs ą 0. Thus, using the cone mixing property, we get: ´PrAsφpLq ď PrA X pθx Aqc s ´ PrAsPrΩ ´ As ď PrAsφpLq (2.10) On the other hand, since E is an invariant set: Prθx A4Es “ Prθx A4θx Es “ Prθx pA4Eqs ă , (2.11) PrA4θx As ď PrpA4Eq Y pθx A4θx Eqs ă 2. (2.12) which implies: In turn, from inequality (2.12), it is clear that PrA X pθx Aqc s ă 2. Now, using the inequality (2.10) one has that PrAsPrΩ ´ As ď 2 ` PrAsφpLq. As a result, the inequalities PrEsPrΩ ´ Es ă pPrAs ` qpPrΩ ´ As ` q (2.13) “ PrAsPrΩ ´ As ` ` 2 (2.14) ă 4 ` φpLq ď 5 (2.15) hold. Hence, from ą 0 was arbitrary this turns out that PrEsPrΩ ´ Es “ 0. Therefore if PrEs ą 0, this implies PrEs “ 1. 2.2.3 Polynomial Decay implies Polynomial decay in a neighborhood In this subsection we prove that whenever pP CqM,c |l holds, for prescribed positive constants M and c, then we can choose 2pd ´ 1q directions where we still have polynomial decay although of less order. More precisely, we can prove the following: Proposition 2.2.3. Suppose that pP qM,c |l is satisfied with c ą 0 for some M ą 6pd ´ 1q, then there exists an α ą 0 such that if we define for i P r2, ds: l`i :“ l ˘ αRpei q |l ` αRpei q| 56 and l´i :“ l ´ αRpei q , |l ´ αRpei q| then pP qN,2c |l˘i is satisfied, where we can choose N “ M 3 ´ 1. Therefore, if M fulfils the prescribed inequality in Theorem 2.2.3, then pP CqM,c |l implies for each i P r2, ds that pP qN,2c |l˘i is satisfied. The loss of degree in the polynomial condition is due to the requirement that the underlying boxes in the condition have the same dimensions in both l and ´l directions. Proof of Proposition 2.2.3. We will just give the proof for direction l´2 , the other cases being analogous. Throughout the proof we pick α P p0, 1q and we define the angle β by: β :“ arctanpαq. Consider the specific rotation R2 on Rd defined ¨ cospβq ´ sinpβq ˚ ˚ ˚ sinpβq cospβq ˚ ˚ ˚ 0 0 ˚ .. .. ˚ 2 R :“ ˚ . . ˚ ˚ .. .. ˚ . . ˚ ˚ ˚ 0 0 ˝ 0 0 (2.16) by: ˛ 0 ... ... 0 0 ... ... 0 1 .. . .. . ... ... .. .. . . .. .. . . 0 .. . .. . ... ... 1 0 ... ... 0 1 ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹. ‹ ‹ ‹ ‹ ‹ ‹ ‚ where this representation matrix is taken in the vector space base tRpe1 q, Rpe2 q, . . . , Rped qu. It will be useful to define a new rotation R1 :“ R2 R rL p0q given by together with the rotated box B ´ ¯ rL p0q :“ R1 r´Lλ1 pαq, Lλ2 pαqs ˆ r´Lcλ3 pαq, Lcλ3 pαqsd´1 X Zd B 57 where: λ1 pαq :“ ? 1 1` α cot2 pβq`1 1 ´1 λ2 pαq :“ ? α2 cot pβq`1 ? p1´cotpβqq2 `p1´tanpβqq2 λ3 pαq :“ | tanpβq`cotpβq| Notice that with these definitions, P0 - almost surely: XTBr L p0q rL p0q ñ XT R B`B R B ` BL,L,l p0q. BL,L,l p0q (2.17) The following figure shows the boxes involved in 2.17. eL (0) B 0 l BL,L,l (0) Figure 2.1: The choice of boxes. As a result we have got P0 rXTBr L p0q rL p0qs ď L´M . R B`B Furthermore, a straightforward computation makes us see that the scale factor λ3 pαq is less than 4 3 whenever α ď 19 . Therefore if we let the positive α ď 4 λ3 pαq ď . 3 1 9 one has that (2.18) For technical reasons, we need to introduce an auxiliary box. Specifically, we first set: 1 α ´1 1´α “? h :“ b 1 ` α2 1 ` p α1 q2 58 and observe that 4 5 ă h ă 1. We then can introduce the new box B l´2 ,L p0q defined by: ´ ¯ B l´2 ,L p0q :“ R1 r´Lph ` 2q, Lhs ˆ r´cLλ3 pαq, Lcλ3 pαqsd´1 X Zd . From this definition, we obtain: P0 rXTB l´2 ,L p0q R B ` B l´2 ,L p0qs ď L´M . In order to complete the proof, we claim that for large enough U the probability: P0 rXTB l´2 ,U p0q R B ` B l´2 ,U p0qs. has polynomial decay on U , where the box B l´2 ,U p0q is defined by B l´2 ,U p0q :“ R1 pr´U, U s ˆ r´2cU, 2cU sd´1 q. The general strategy to follow will be to stack smaller boxes up inside of B l´2 ,U p0q and then using the Markov property along with good environment sets we will ensure that the walk exits from box B l´2 ,U p0q by B ` B l´2 ,U p0q with probability bigger than 1 ´ 1{P pU q, where P is a polynomial function. Specifically, we let L :“ U . h`2 (2.19) We introduce a sequence of stopping times as follows: T1 “ TB l ´2 ,L p0q , and for i ą 1 Ti “ Ti´1 ` T1 ˝ θTi´1 . For simplicity we write Tp1 instead of TB l ´2 ,U p0q . In view of (2.18) and (2.19) it is clear that four successful exits of the walk from boxes of the B l´2 ,L p0q-type are sufficient to ensure that the walk exits from B l´2 ,U p0q by its positive boundary. Therefore one sees that “ P0 rXTx1 P B ` B l´2 ,U p0qs ě P0 XT1 P B ` B l´2 ,L p0q, ` ˘ ` ˘ XT1 P B ` B l´2 ,L pXT1 q ˝ θT1 , XT1 P B ` B l´2 ,L pXT2 q ˝ θT2 , ` ˘ ‰ XT1 P B ` B l´2 ,L pXT3 q ˝ θT3 59 (2.20) In order to use (2.20), let i be a positive integer number and consider the lattice sets sequence pFi qiě1 defined by: F1 “ B ` B l´2 ,L p0q, and for i ą 1, we define by induction: Fi “ ď B ` B l´2 ,L pyq. yPF1 We now define for i ě 1, the environment events Gi by: ) ` ˘ M Gi “ ω P Ω : Py,ω r XT1 P B ` B l´2 ,L pXTi q ˝ θTi s ě 1 ´ L´ 2 , for each y P Fi Plainly it is satisfied P0 rXTx1 P B ` B l´2 ,U p0qs ě “ ` ˘ P0 XT1 P B ` B l´2 ,L p0q, XT1 P B ` B l´2 ,L pXT1 q ˝ θT1 , ` ˘ ˘ ` ‰ XT1 P B ` B l´2 ,L pXT2 q ˝ θT2 , XT1 P B ` B l´2 ,L pXT3 q ˝ θT3 ě ` ˘ “ P0 XT1 P B ` B l´2 ,L p0q, XT1 P B ` B l´2 ,L pXT1 q ˝ θT1 , ˘ ˘ ` ` ‰ XT1 P B ` B l´2 ,L pXT2 q ˝ θT2 , XT1 P B ` B l´2 ,L pXT3 q ˝ θT3 1G3 By the Markov property applied at time T3 and the very meaning of G3 , we get that the last expression equals: “ “ ` ˘ ` ` B p0q, X B pX P B q ˝ θT1 , E P X P B 0,ω T l ,L T l ,L T ´2 1 ´2 1 1 yPF3 ˘ ‰ ` ` ` P B B l´2 ,L pyqs1G3 ě XT1 P B B l´2 ,L pXT2 q ˝ θT2 Py,ω rXTB l´2 ,L pyq “ ` ˘ M ` p1 ´ L´ 2 q P0 XT1 P B ` B l´2 ,L p0q, XT1 P B ` B l´2 ,L pXT1 q ˝ θT1 , ` ˘ ‰ ˘ XT1 P B ` B l´2 ,L pXT2 q ˝ θT2 ´ PrpG3 qc s ř . (2.21) (2.22) Repeating the above argument, one has the following upper bound for the right most expression of (2.21): M M M M p1 ´ L´ 2 q4 ´ p1 ´ L´ 2 q3 PrpG1 qc s ´ p1 ´ L´ 2 q2 PrpG2 qc s ´ p1 ´ L´ 2 qPrpG3 qc s. (2.23) 60 At this point, we would like to obtain for i P |r1, 3s|, an upper bound of the probabilities: PrpGi qc s. To this end, we first observe that Chevyshev’s inequality and our hypothesis imply: PrpG1 qc s ď ÿ yPF1 M Er1tP y,ω rXT Bl Clearly, we have the estimate | F1 |ď ´2 ,L pyq PB ` B l´2 ,L pyqsąL´ M 2 u s ď| F1 | L´ 2 . ` 8 ˘d´1 L (recall (2.18)). As a result, we have that: 3 ˆ c PrpG1 q s ď ˙d´1 M 8 L L´ 2 . 3 (2.24) By a similar procedure we can conclude that ˙d´1 ˆ M 16 c L L´ 2 . PrpG2 q s ď 3 and ˆ c PrpG3 q s ď 24 L 3 (2.25) ˙d´1 M L´ 2 . (2.26) Combining the estimates in (2.20)-(2.26) and the assumption M ě 6pd ´ 1q we see that: P0 rXTx1 R B ` B l´2 ,U p0qs ď 3 6p8qd´1 ´M 3 M U´ 3 . 2 This ends the proof by choosing the required α as any number in the open interval p0, 91 q. 2.3 Examples of directionally transient random walks without an asymptotic direction and vanishing velocity We will present two examples of random walks in random environment which exhibit the possible limitations of the hypothesis of a theorem stating the existence of an asymptotic direction and of a theorem stating the existence of a non-vanishing velocity for mixing environments. 61 Assumption TNB. Let p be a random variable taking values in p0, 1q such that there exists a unique κ P p1{2, 1q with the property that Erρκ s “ 1 and Erρκ ln` ρs ă 8, where ρ :“ p1 ´ pq{p. 2.3.1 Random walk with a vanishing velocity but with an asymptotic direction Let tpi : i P Zu be i.i.d. copies of p. Let e1 and e2 be the canonical vectors in Z2 . Define an i.i.d. sequence of random variables tωi : i P Zu with ωi “ tωi pe1 q, ωi p´e1 q, ωi pe2 q, ωi p´e2 qu, by 1 ωi pe2 q “ ωi p´e2 q “ , 4 ωi pe1 q “ pi 2 and ωi p´e1 q “ 1 pi ´ . 2 2 Now consider the random environment ω “ tωppi, jqq : pi, jq P Z2 u defined ωppi, jqq :“ ωi for all i, j P Z. We will call P1 the law of the above environment and Q1 the annealed law of the corresponding random walk starting from 0. Theorem 2.3.1. Consider a random walk in a random environment with law P1 . Then, the following are satisfied: piq Q1 -a.s. lim Xn ¨ e1 “ 8. nÑ8 62 piiq Q1 -a.s. Xn “ 0. nÑ8 n lim piiiq In Q1 -probability Xn “ e1 . nÑ8 |Xn |2 lim pivq The law Q1 satisfies the polynomial condition pP CqM,c is satisfied, with M “ κ´ 12 ´ε and c “ 1, where ε is an arbitrary number in the interval p0, κ ´ 21 q. Proof. piq We will describe a one dimensional procedure which will be used throughout the proofs of items (i) and piiq. Specifically, defining pYn qně0 :“ pXn ¨ ei qně0 one has that it can be identified with the one dimensional RWRE which has quenched law P0,ω starting from 0, defined by the transition probabilities: r pYn , e1 q “ pYn {2, P0,ω rYn`1 “ Yn ` e1 | Yn s “ ω P0,ω rYn`1 “ Yn ´ e1 | Yn s “ ω r pYn , ´e1 q “ p1 ´ pYn q{2, and r pYn , 0q “ 1{2. P0,ω rYn`1 “ Yn | Yn s “ ω r1 rlnrρr0 ss ă 0, where ρr0 :“ ω r p0, ´e1 q{r Since assumption TNB it follows that E ω p0, e1 q r1 denotes the corresponding expectation in this random environment. Now, and E from the transience criteria in [Ze1] Theorem 2.1.2 one has that Q1 - a.s. lim Xn ¨ e1 “ 8. nÑ8 piiq Since κ ď 1, using a one dimensional procedure for directions e1 and e2 and the strong law of large numbers for one dimensional RWRE ([Ze1], Theorem 2.1.9), we get Q1 -a.s. Xn pXn ¨ e1 qe1 ` pXn ¨ e2 qe2 “ Ñ 0. n n 63 piiiq We define the random variables N1 and N2 as horizontal and vertical steps performed by the walk Xn , respectively. By the very definition of this example, both of them distribute Bpn, 12 q under the quenched law. Given ε ą 0, we have to estimate the following probability: ˇ »ˇ fi ˇ «ˇ ff ˇ pXn ¨e1 q ˇ pXn ¨e2 q ˇ pX ¨ e qe ` pX ¨ e qe ˇ ˇ κ e1 ` nκ e2 ˇ ˇ n 1 1 ˇ n 2 2 Q1 ˇ a ´ e1 ˇ ą ε “ Q1 –ˇˇ bn ´ e1 ˇˇ ą εfl . ˇ pXn ¨ e1 q2 ` pXn ¨ e2 q2 ˇ ˇ pXn ¨e1 q2 ` pXn ¨e2 q2 ˇ n2κ n2κ Clearly, Xn ¨ e2 under the annealed law has the same law Pr of the one dimensional simple symmetric random walk Zn1 at time n1 “ N2 , such that Pr- a.s. n1 {n Ñ 1{2 as n Ñ 8. Therefore, since κ ą 1{2 as a result one has „ „ 1 Xn ¨ e2 Z N 2 “ 0 “ Pr lim “ 0 “ 1. Q1 lim nÑ8 N2 Ñ8 N2κ 2κ nκ and also Q1 - a.s. pXn ¨ e2 q2 “ 0. nÑ8 n2κ lim On the other hand, using the convergence theorem of Kesten, Kozlov and Spitzer [KKS75], calling YN1 the one dimensional random walk in random environment corresponding to Xn ¨ e1 and using a similar procedure as the one given above, we can see that a Xn ¨ e1 Ñ1 pXn ¨ e1 q2 in distribution, which turns out that this convergence is also in Q1 - probability and completes the proof. e pivq For j P t1, 2u and a a positive real number, we define the stopping times Ta j and e Tra j by Taej :“ inftn ě 0 : Xn ¨ ej ě au (2.27) Traej :“ inftn ě 0 : Xn ¨ ej ď au (2.28) along with Notice that for c “ 1 and large L one has the following estimate e1 e2 Q1 rXTBL,cL,l p0q R B ` BL,cL,l p0qs ď Q1 rTr´L ă TLe1 s ` Q1 rTLe2 ^ Tr´L ă TLe1 s. 64 (2.29) The first probability in the right most side of (2.29) has an exponential bound as it follows from the estimate in the proof of item pivq in Theorem 2.3.2. Observe that the second probability in the right most side of (2.29) is less than or equal to e2 Q1 rTLe2 ^ Tr´L ď L2`ε s ` Q1 rL2`ε ă TLe1 s. Keeping the notations introduced in item piiiq. From the very definition of Zn1 , one sees that for large L, there exists a positive constant K1 such that: e2 Q1 rTLe2 ^ Tr´L ď L2`ε s ď Q1 r|Xn ¨ e2 | ď L, for all n P N, 0 ď n ď L2`ε s ď PrrZN2 pnq ď L2`ε , for all n P N, 0 ď n ď L2`ε s ď expt´K1 Lε u. (2.30) On the other hand, using the sharp estimate in Theorem 1.3 in [FGP10] and denoting P̂ the law of underlying one dimensional random walk corresponding to the annealed law of pXn ¨ e1 qně0 , we can see that for large L, there exists a positive constant K2 such that: Q1 rL2`ε ă TLe1 s ď Q1 rXrL2`ε s ¨ e1 ă Ls ď P̂ rYN1 prL2`ε sq ă Ls ď K2 L´pκ´1{2´εq . (2.31) (2.32) Therefore, in view of the inequality (2.29) and the estimates (2.30)-(2.31), we complete the proof. 2.3.2 Directionally transient random walk without an asymptotic direction Let tpi : i P Zu and tp1j : j P Zu be two independent i.i.d. copies of p. Following a similar procedure as in the previous example, we consider in the lattice Z2 the canonical vectors e1 and e2 , and define the random environment ω “ tωppi, jqq : pi, jq P Z2 u by, 65 ωpi,jq pe1 q “ pi 2 and ωpi,jq p´e1 q “ 1 pi ´ . 2 2 ωpi,jq pe2 q “ p1j 2 and ωpi,jq p´e2 q “ 1 p1j ´ . 2 2 together with We call P2 the law of the above environment and Q2 the annealed law of the corresponding random walk starting from 0. Theorem 2.3.2. Consider a random walk in a random environment with law P2 . Then, the following are satisfied. piq Let l P S. Then l ¨ e1 ě 0 and l ¨ e2 ě 0 if and only if Q2 -a.s. lim Xn ¨ l “ 8. nÑ8 piiq Q2 -a.s. Xn “ 0. nÑ8 n lim piiiq There exists a non-deterministic v̂ such that: Xn Ñ v̂. |Xn |2 in distribution. pivq There exists a c ą 1 such that limLÑ8 L´1 log Q2 rXTBL,cL,l p0q R B ` BL,cL,l p0qs ă 0, ? ? where l “ p1{ 2, 1{ 2q. Thus, condition pT q|l [Sz02] is satisfied. Proof. 66 (2.33) piq This amounts to prove that Q2 - a.s. lim Xn ¨ e1 “ 8 and lim Xn ¨ e2 “ 8. nÑ8 nÑ8 Both assertions follow from the one dimensional procedure, Theorem 2.1.2 in [Ze1] and the assumption TNB. piiq This proof is similar to case piiq of Theorem 2.3.1 . piiiq Define a sequence Ti,j , for i ě 0, j P t1, 2u as follows: T0,j “ 0, T1,j “ inftn ě 0 : pXn ´ X0 q ¨ ej ą 0 or pXn ´ X0 q ¨ ej ă 0u and for i ě 2 Ti,j “ T1,j ˝ θTi´1,j ` Ti´1,j . Setting Yn,j :“ XTn,j ¨ ej , we see that for j P t1, 2u, the one dimensional random walks without transitions to itself at each site pYn,j qně0 are independent and their transitions at each site i P Zd are determined by pi . Furthermore, for j P t1, 2u, the strong law of large numbers implies that Q2 - a.s. Tn,j “ 2. nÑ8 n (2.34) lim We now apply the result of Kesten, Kozlov and Spitzer [KKS75] to see that there exist constants C1 and C2 such that ˆ ˙ ˆ ˆ ˙κ ˆ ˙κ ˙ Yn,1 Yn,2 1 1 Ñ C1 , , C2 1 κ 2 κ nκ nκ Sca Sca j κ in distribution, where for j P t1, 2u, Sca stand for two independent completely asymmetric stable laws of index κ, which are positive. Using (2.34) and properties of convergence in distribution we can see that ´ ¯κ ´ ¯κ C1 C2 pXn ¨e1 q pXn ¨e2 q e ` e2 1 1 κ 2 κ e1 ` nκ e2 Sca Sca Xn κ “ bn Ñ c´ ¯2κ ´ ¯2κ |Xn | pXn ¨e1 q2 pXn ¨e2 q2 C2 C1 ` 2κ 2κ ` S2 κ n n S1 κ ca ca in distribution. Therefore we have proved that the limit v̂ is random. 67 pivq A first step will be to prove the following decay e e lim sup L´1 log Q2 rTr´rjcL ă TcLj s ă 0 for arbitrary positive constants r c and c (see (2.27) and (2.28) for the notations). We prove this in the case j “ 1, the another case being similar. Following the notation introduced in Theorem 2.3.1 item piq and denoting the greatest integer function by r¨s, we see that it is sufficient to prove that for large L there exists a positive p such that: constant C r1 rP0,ω rYn hits rp p E cLs ` 1 before rcLs ` 1ss ď expt´CLu. (2.35) To this end, for a fixed random environment ω, if we denote VLi to Pi,ω rYn hits ´ rp cLs ` 1 before rcLs ` 1s, the Markov property makes us see that VLi satisfies the following difference equation for integer i P rr cLs ` 2, rcLss VLi “ p1 ´ pi qVLi´1 ` p1 VLi`1 with the constraints VLrrcLs`1 “ 1 and VLrcLs`1 “ 0. This system can be solved by the method developed by Chung in [Ch67], Chapter 1, Section 12. Applying this one sees that ř ř expt u ` . . . ` expt c L`1s,0 cLs`1,rcLs u ´rp ´rp ř ř , VL0 “ 1 ` expt ´rpcLs`1,´rpcLs`2 u ` . . . ` expt ´rpcLs`1,rcLs u ř where we have adopted the notation introduced in [Sz02], zămďz1 “ log ρpmq and ρpmq “ p1´pm q{pm . A slight variation of the argument in [Sz02] page 744 completes the proof of claim (2.35). On the other hand, considering the probability Q2 rXTBL,2L,l p0q R B ` BL,2L,l p0qs, we observe that is clearly bounded from above by (see Figure 2.2) Q2 rTre1?2 ă T?e12L s ` Q2 rTre2?2 ă T?e22L s ´ 2 L ´ 2 L In virtue of the claim (2.35) the last expression has an exponential bound and this finishes the proof. 68 √1 , √1 2 2 L Slab H2 2L Slab H1 Figure 2.2: A geometric sketch to bound Q2 rXTBL,2L,l p0q R B ` BL,2L,l p0qs. 2.4 Backtracking of the random walk out of a cone Here we will provide a uniform control on the probability that a random walk starting form the vertex of a cone stays inside the cone forever. It will be useful to this end to define D1 :“ inftn P N : Xn R Cpα, l, X0 qu, (2.36) where as before l P Sd´1 . Proposition 2.4.1. Let l P Sd´1 . Suppose that pP qM,c |l holds, for some M ą 6d ´ 3. Then there exists a positive constant c2 pdq ą 0 such that P0 rD1 “ 8s ą c2 pdq. In what follows we prove this proposition. With the purpose of making easier the reading, we introduce here some notations. Let l1 P Sd´1 and choose a rotation R1 on Rd with the property R1 pe1 q “ l1 For each x P Zd , real numbers m ą 0, c ą 0 and integer i ě 0 we define the box Bi pxq :“ ´ ¯ d´1 x ` R1 p´2m`i , 2m`i q ˆ p´2c2m`i , 2c2m`i q X Zd 69 along with its ”positive boundary” ` ˘ B ` Bi pxq :“ BBi pxq X tx ` R1 p2m`i , 8q ˆ Rd´1 u. We also need slabs perpendicular to direction l1 . Set ` ˘ V0 pxq :“ x ` R1 r´2m , 2m s ˆ Rd´1 X Zd and for i ě 1, ˜« ´2m , Vi pxq :“ x ` R1 i ÿ ff ¸ 2m`j ˆ Rd´1 X Zd . j“0 The positive part of the boundary for this set is defined as # ˜˜ B ` Vi pxq :“ BVi pxq X x ` R1 i ÿ ¸ 2m`j , 8 ¸+ ˆ Rd´1 . j“0 Furthermore, we will define recursively a sequence of stopping times as follows. First, let T0 :“ TB0 pX0 q . and for i ě 1 Ti :“ TBi pXTi´1 q ˝ θTi´1 ` Ti´1 . We now need to define the first time of entrance of the random walk to the hyperplane ` ˘ R1 p´8, 0q ˆ Rd´1 , Dl1 :“ inftn ě 0 : Xn ¨ l1 ă 0u. With these notations we can prove: Lemma 2.4.2. Assume pP qN,2c |l1 where c ą 0 , for some N ą 2pd ´ 1q. Then, for all m P N and x P tz P Zd : z ¨ l1 ě 2m u, we have that Px rDl1 “ 8s ě ypmq where ypmq does not depend on l1 and satisfies limmÑ8 ypmq “ 1. 70 Proof. From the fact that pP qN,2 |l1 holds, we can (and we do) assume that there exists a m ą 0 large enough, such that for any positive integer i one has that P0 rXTBi p0q P B ` Bi p0qs ě 1 ´ 2´N pm`iq (2.37) holds. By stationarity, we have for x P Zd : Px rXTBi pxq P B ` Bi pxqs ě 1 ´ 2´N pm`iq . (2.38) Throughout this proof, let us choose x P tz P Zd : z ¨ l1 ě 2m u. For reasons that will be clear through the proof, we need to estimate for i ě 1 the following probability Ii :“ Px rXTVi pxq P B ` Vi pxqs, (2.39) and with this aim, in view of (2.38), we have m I0 ě Px rXTB0 pxq P B ` B0 pxqs ě 1 ´ 2´N m ě 1 ´ 2´N 2 . Now, as a preliminary computation for the recursion, we begin to estimate I1 . Note that I1 ě Px rXT0 P B ` B0 pX0 q, pXTB1 pX0 q P B ` B1 pX0 qq ˝ θT0 s. (2.40) Using the strong Markov property at time T0 we then see that I1 ě ě ř ř E rPx,ω rXT0 P B ` B0 pX0 q, XT0 “ ys ı ˆ Py,ω rXTB1 pyq P B ` B1 pyqs yPB ` B0 pxq E rPx,ω rXT0 P B ` B0 pX0 q, XT0 “ ys ı ˆ Py,ω rXTB1 pyq P B ` B1 pyqs1G0 , yPB ` B0 pxq where G0 :“ m tw P Ω : Py,ω rXTB1 pyq P B ` B1 pyqs ą 1 ´ 2´N 2 , for all y P B ` B0 pxqu. 71 (2.41) Thus, it is clear that ˘ ` m˘` I1 ě 1 ´ 2´N 2 Px rXT0 P B ` B0 pX0 qs ´ PrpG0 qc s . (2.42) Notice that by (2.38) and Chebyshev’s inequality PrpG0 qc s ď ř ď ř m yPB ` B0 pxq PrPy,ω rXTB1 pyq R B ` B1 pyqs ě 2´N 2 s yPB ` B0 pxq Py rXTB1 pyq R B ` B1 pyqs2N 2 m m “ |B ` B0 pxq|2N 2 sP0 rXTB1 p0q R B ` B1 p0qs m m ď p2c2m`1 qd´1 2N p 2 ´pm`1qq ď p2c2m`1 qd´1 2´N 2 . (2.43) Plugging (2.43) into (2.42) we see that m m m I1 ě p1 ´ 2´N 2 qp1 ´ 2´N 2 ´ p2c2m`1 qd´1 2´N 2 q. (2.44) Hereafter we can do the general recursive procedure. For this end, we define for i ě 1 Ji :“ P0 rXT0 P B ` B0 pX0 q, pXTB1 pX0 q P B ` B1 pX0 qq ˝ θT0 , . . . (2.45) ` . . . , pXTBi pX0 q P B Bi pX0 qq ˝ θTi´1 s. It is straightforward that Ii ě Ji . Furthermore, through induction on i ě 1, we will establish the following claim » Ji ě p1 ´ 2´N pm`i´1q 2 ˜ q –Ji´1 ´ 2´N pm`i´1q 2 i´1 ÿ ¸d´1 fi 2c2pm`jq`1 fl . (2.46) j“0 To prove this, we first define the extended boundary of the pile of boxes at a given step as F0 :“ BB0 pxq X tx ` R1 pp2m , 8q ˆ Rd´1 qqu, and for i ě 2 ( Fi´1 :“ B YyPFi´2 Bi´1 pyq X tx ` R1 pp2m`i´1 , 8q ˆ Rd´1 qqu. Using these notations, we can apply the strong Markov property to (2.45) at time Ti´1 , to get that Ji “ ř yPFi´1 E rPx,ω rXT0 P B ` B0 pX0 q, . . . ı . . . , pXTBi´1 pX0 q P B ` Bi´1 pX0 qq ˝ θTi´2 , XTi´1 “ ysPy,ω rXTBi pX0 q P B ` Bi pX0 qs . 72 Following the same strategy used to deduce (2.44), it will be convenient to introduce for each i ě 2 the event Gi´1 :“ tω P Ω : Py,ω rXTBi pyq P B ` Bi pyqs ą 1 ´ 2´N pm`i´1q 2 , for all y P Fi´1 u. Inserting the indicator function of the event Gi´1 into (2.45) we get that Ji ě ” ř ` ` yPFi´1 E Px,ω rXT0 P B B0 pX0 q, . . . , pXTBi´1 pX0 q P B Bi´1 pX0 qq ˝ θTi´2 , XTi´1 “ ys ı ` ˆPy,ω rXTBi pX0 q P B Bi pX0 qs1Gi´1 . By the same kind of estimation as in (2.42), we have Ji ě p1 ´ 2´N pm`i´1q 2 q pJi´1 ´ PrpGi´1 qc sq . (2.47) We need to get an estimate for PrpGi´1 qc s. We do it repeating the argument given in (2.43). Let us first remark that ˜ i´1 ÿ |Fi´1 | ď ¸d´1 2c2pm`jq`1 , (2.48) j“0 holds. Indeed, the case in which l1 “ e1 gives the maximum number for |Fi´1 |. Keeping (2.48) in mind we get that ı ” ” ı ´N pm`i´1q ` 2 P P X R B B pyq ě 2 i y,ω T Bi pyq yPFi´1 Px rpGi´1 qc s ď ř ď ř pm`i´1q Py rXTBi pyq R B ` Bi pyqs2N 2 ´ř ¯d´1 pm`i´1q i´1 pm`jq`1 ď 2´N 2 . 2c2 j“0 yPFi´1 (2.49) Therefore, combining (2.49) and (2.47) we prove claim (2.46). Iterating (2.46) backward, from a given integer i, we have got « ff i´1 i´1 i´1 ź ÿ pm`hq pm`kq m`j ź Ji ě J1 p1 ´ 2´N 2 q ´ aj 2´N 2 p1 ´ 2´N 2 q, j“1 h“1 k“j where we have used for short ˜ j ¸d´1 ÿ aj :“ c2pm`iq`1 ď p2cqd´1 2pm`j`2qpd´1q . i“0 73 (2.50) The same argument used to derive (2.44) can be repeated to conclude that m m m J1 ě p1 ´ 2´N 2 qp1 ´ 2´N 2 ´ p2c2m`1 qd´1 2´N 2 q. (2.51) Replacing the right hand side of (2.51) into (2.50), and together to the fact Ii ě Ji , we see that « Ii ě i´1 ź ff ´N p1 ´ 2 m`h 2 ´N m 2 q p1 ´ 2 i´1 ÿ q´ ´N pm`jq 2 aj 2 j“0 h“0 i´1 ź pm`kq p1 ´ 2´N 2 q. (2.52) k“j Now we can finish the proof. First, observe that Px rDl1 “ 8s ě I8 , where as a matter of definition I8 :“ lim Ii iÑ8 (this limit exists, because it is the limit of a decreasing sequence of real numbers bounded from below). By the condition N ą 2pd ´ 1q, we get that for each m ě 1 one has that for all j ě 1, aj 2´ M pm`jq 2 ď p8cqd´1 2´ϑ pm`jq 2 , where ϑ stands for the positive number so that N “ 2pd ´ 1q ` ϑ. Thus all the products and series in (2.52) converge and we have that for all m ě 1 and x P tz P Zd : z ¨ l1 ě 2m u Px rDl1 “ 8s ě ypmq, where ”ś ı m 8 ´N pm`hq 2 ypmq :“ p1 ´ 2´N 2 q p1 ´ 2 q h“0 ř ś8 ´N pm`jq ´N pm`kq 2 2 ´ 8 q. j“0 aj 2 k“j p1 ´ 2 Clearly for each m ě 1, ypmq does not depend on the direction l1 and limmÑ8 ypmq “ 1, which completes the proof. 74 With the previous Lemma, we now have enough tools to prove Proposition 2.4.1. Before this, we need a definition of geometrical nature. We will say that a sequence px0 , . . . , xn q of lattice points is a path if for every 1 ď i ď n ´ 1, one has that xi and xi´1 are nearest neighbors. Furthermore, we say that this path is admissible if for every 1 ď i ď n ´ 1 one has that pxi ´ xi´1 q ¨ l ‰ 0. Proof of Proposition 2.4.1. Assume pP qM,c |l, where M ą 6pd ´ 1q ` 3 which is the condition of the statement of the Proposition 2.4.1. We appeal to Proposition (2.2.3) and assumption pP qM,c |l to choose an α ą 0 such that for all i P r2, ds pP qN,2c |l˘i is satisfied with N :“ M ´ 1 ą 2pd ´ 1q. 3 (2.53) From now on, let m be any natural number satisfying ypmq ą 1 ´ 1 , 2pd ´ 1q (2.54) where ypmq is the function given in Lemma 2.4.2. Note that there exists a constant c3 pdq such that for all x P Zd contained in Cpα, l, Rp2m e1 qq and such that |Rp2m e1 q ´ x|1 ď 1 one has that there exists an admissible path with at most c3 2m lattice points joining 0 and x. We denote this path by p0, y1 , . . . , yn “ xq noting that n ď c3 2m . The general idea to finish the proof is to push forward the walk up to site x with the help of uniform ellipticity in direction l and then we make use of Lemma (2.4.2) to ensure that the walk remains inside the cone. 75 Therefore, by (2.53) and Lemma (2.4.2) we can conclude that for all 2 ď i ď d one has that Px rDli` “ 8s ě ypmq, (2.55) Px rDli´ “ 8s ě ypmq. (2.56) along with Define the event that the random walk starting from 0 following that path p0, y1 , . . . , yn q as An :“ tpX0 , . . . , Xn q “ p0, y1 , . . . , yn qu. Now notice that P0 rD1 “ 8s ě “ ‰ P0 An , pDli´ “ 8q ˝ θn , p Dli` “ 8q ˝ θn for 2 ď i ď d . (2.57) On the other hand, by definition of the annealed law, together with the strong Markov property we have that P0 rAn , pDli´ “ 8q ˝ θn , p Dli` “ 8q ˝ θn for 2 ď i ď ds “ ‰ “ E P0,ω rAn s, Px,ω rDli´ “ 8, Dli` “ 8 for 2 ď i ď ds . (2.58) Using the uniform ellipticity assumption pU Eq|l, along with (2.55) and (2.56), we can see that (2.58) is bounded from below by m p2κqc3 2 p1 ´ 2pd ´ 1qp1 ´ ypmqqq . (2.59) By virtue of our choice of m in (2.54), we see that there exists a constant c2 just depending on the dimension (we recall that m is fixed at this point of the proof), such that m c4 :“ p2κqc3 2 p1 ´ 2pd ´ 1qp1 ´ ypmqqq ą 0 Finally, in view of the inequalities (2.57) and (2.58) it follows that P0 rD1 “ 8s ě c4 . 76 (2.60) 2.5 Polynomial control of regeneration positions In this section, we define an approximate regeneration times as done in [CZ01], which will depend on a distance parameter L ą 0. We will then show that these times, assuming pP CqM,c |l for M large enough, and cone-mixing, when scaled by κL , define approximate regeneration positions with a finite second moment. 2.5.1 Preliminaries We recall the definition of approximate renewal time given in [CZ01]. Let W :“ E Y t0u [c.f. (2.2)] and endow the space W N with the canonical σ´algebra W generated by the cylinder sets. For fixed ω P Ω and ε “ pε0 , ε1 , . . .q P W N , we denote by Pω,ε the law of the Markov chain tXn u on pZd qN , so that X0 “ 0 and with transition probabilities defined for z P Zd , e, |e| “ 1 as Pω,ε rXn`1 “ z ` e|Xn “ zs “ 1tεn “eu ` 1tεn “0u rωpz, eq ´ κ1tePEu s. 1 ´ κ|E| Call Eω, the corresponding expectation. Define also the product measure Q, which to each sequence of the form ε P W N assigns the probability Qpε1 “ eq :“ κ, if e P E, while Qpε1 “ 0q “ 1 ´ κ|E|, and denote by EQ the corresponding expectation. Now let G be the σ-algebra on pZd qN generated by cylinder sets, while F be the σalgebra on Ω generated by cylinder sets. Then, we can define for fixed ω the measure P 0,ω :“ Q b Pω,ε on the space pW N ˆ pZd qN , W ˆ Gq, and also P 0 :“ P b Q b Pω,ε on pΩˆW N ˆpZd qN , FˆWˆGq, denoting by Ē0,ω and Ē0 the corresponding expectations. A straightforward computation makes us conclude that the law of tXn u under P̄0,ω coincides with its law under P0,ω and that its law under P 0 coincides with its law under P0 . Let q be a positive real number such that for all 1 ď i ď d, 77 ui :“ li q is an integer. Define now the vector u :“ pu1 , . . . , ud q. From now on, we fix a particular sequence ε in E of length p :“ |u|1 whose components sum up to u: ε :“ pε1 , . . . , εp q, together with ε1 “ ε2 “ . . . “ ε|u1 | :“ sgnpu1 qe1 , ε|u1 |`1 “ ε|u1 |`2 “ . . . “ ε|u1 |`|u2 | :“ sgnpu2 qe2 .. . εp´|ud |`1 “ . . . “ εp :“ sgnpud qed . Without loss of generality we can assume that l1 “ 0. And by taking α small enough that ε1 , ε1 ` ε2 , . . . εp are inside of Cp0, l, αq. For L P pN consider the sequence ε̄pLq of length L, defined as the concatenation L{p times with itself of the sequence ε̄, so that εpLq “ pε̄1 , . . . , ε̄p , . . . , ε̄1 , . . . , ε̄p q. Consider the filtration G :“ tGn : n ě 0u where Gn :“ σppεi , Xi q, i ď nq. Define S0 :“ 0, S1 :“ inftn ě L : Xn´L ¨ l ą maxtXm ¨ l : m ă n ´ Lu, pεn´L , . . . , εn´1 q “ εpLq u together with R1 :“ D1 ˝ θS1 ` S1 . 78 We can now recursively define for k ě 1, Sk`1 :“ inftn ě Rk : Xn´L ¨ l ą maxtXm ¨ l : m ă n ´ Lu, pεn´L , . . . , εn´1 “ εpLq qqu and Rk`1 :“ D1 ˝ θSk`1 ` Sk`1 . Clearly, 0 “ S0 ď S1 ď R1 ď ¨ ¨ ¨ 8, the inequalities are strict if the left member of the corresponding inequality is finite, and the sequences tSk : k ě 0u and tRk : k ě 0u are G-stopping times. On the other hand, we can check that P̄0 ´a.s. one has that S1 ă 8 along with the fact P̄0 ´a.s. on the set tlim Xn ¨ l “ 8u X tRk ă 8u one has too that (2.61) Sk`1 ă 8. Put K :“ inftk ě 1 : Sk ă 8, Rk “ 8u and define the approximate regeneration time τ pLq :“ SK . (2.62) We see that the random variable τ pLq is the first time n in which the walk has reached a record at time n ´ L in direction l, and then the walk goes on L steps in the direction l by means of the action of εpLq to finally after this time n never exits the cone CpXn , l, αq. The following lemma is required to show that the approximate renewal times are P̄0 a.s. finite. Its can be proved using a slight variation of the argument given in page 517 of Sznitman [Sz03]. Lemma 2.5.1. Consider a random walk in a random environment. Let l P S˚ d´1 , M ě d ` 1 and c ą 0 and assume that pP CqM,c |l is satisfied. Then the random walk is transient in direction l. 79 Proof. For the sake of completeness, we are going to sketch the steps so as to obtain the claim of the theorem. Step 1. Notice that any M ą 0 gives P0 rlim sup Xn ¨ l “ 8s “ 1. nÑ8 From one can easily show that P0 rlim sup Xn ¨ l “ 8s “ 1 nÑ8 if and only if lim P0 rinftn ě 0 : Xn ¨ l ě Lu “ 8s “ 0. LÑ8 Step 2. Following the argument on page 517 of [Sz03], we have got to get rid the order of the positive boundary of a box plus some order which makes possible to apply BorelCantelli Lemma. It can be seen that a term in M of d ´ 1 suffices to get rid the order of the positive boundary and therefore M ě pd ´ 1q ` 2 “ d ` 1 is enough to get: P0 r lim Xn ¨ l “ 8s “ 1. nÑ8 We make note a trivial remark that the random walk is transient in direction u also. We can now prove the following stronger version of Lemma 2.2 of [CZ01]. Lemma 2.5.2. Assume pCM qα,φ |l, pU Eq|l and pP CqM,c for M ą 6d ´ 3, c ą 0. Then there exists a positive L0 P |u|1 N, such that φpL0 q ` P0 rD1 ă 8s ă 1, and τ pLq ă 8, P0 -a.s. are fulfilled for each L ě L0 , L P |u|1 N. Proof. Following the arguments in the proof of Lemma 2.2. of [CZ01] (using u instead of l), one has that: P̄0 rRk ă 8s ď pφpL0 q ` P0 rD1 ă 8sqk 80 (2.63) From the assumption pCM qα,φ |l, we have φpLq Ñ 0 as L Ñ 8. On the other hand, by Lemma 2.4.1, P0 rD1 ă 8s ă 1. Therefore, we can find a L0 with the property: φpLq ` P0 rD1 ă 8s ă 1, for all L ě L0 , L P N|l|1 . Then, via Borel-Cantelli Lemma, one has that P̄0 ´ almost surely inftn ě 1 : Rn “ 8u ă 8, (2.64) holds. Now, observe that P̄0 ´ almost surely: inftn ě 1 : Rn “ 8u “ inftn ě 1 : Rn´1 ă 8 Rn “ 8u (2.65) In turn, using (2.61) which is satisfied in view of Lemma 2.5.1, turns out that inftn ě 1 : Sn ă 8 Rn “ 8u “ K ă 8 P̄0 ´ almost surely. Finally, we can state the following proposition, which gives a control on the second moment of the position of the random walk at the first regeneration position. Define for x P Zd and L ą 0 the σ´algebra " Fx,L * L :“ σ ωpy, ¨q; y ¨ u ď x ¨ u ´ |u|2 . |u|1 Proposition 2.5.3. Fix l P S˚ d´1 , α ą 0, M ą 0 and φ : r0, 8q Ñ r0, 8q such that 1 limrÑ8 φprq “ 0. Assume that 0 ă α ă mint 19 , 2c`1 u and that pCM qα,φ |l, pU Eq|l and pP CqM,c |l hold. Then, there exists a constant c5 , such that Ē0 rpκL Xτ pLq ¨ lq2 |F0,L s ď c5 . 81 (2.66) 2.5.2 Preparatory results Now we are in position to prove the main proposition of this section. Before we do this, we will prove a couple of lemmas. Lemma 2.5.4. Assume that pCM qα,φ |l holds. Then, for each x P Zd one has that |ErPx,ω rD1 “ 8s|Fx,L s ´ P0 rD1 “ 8s| ď φpLq holds a.s. Proof. For each A P Fx,L , we define νrAs :“ ErPx,ω rD1 “ 8s1A s (2.67) µrAs :“ pP0 rD1 “ 8s ` φpLqq PrAs ´ νrAs. (2.68) and Clearly (2.67) defines a measure on pΩ, Fx,L q. We will show that (2.68) also. Indeed, take an A P Fx,L and note that Px,ω rD “ 8s is σtωpy, ¨q, y P Cpx, l, αqu-measurable. Therefore, by assumption pCM qα,φ |l one has that νrAs ď P0 rD1 “ 8sPrAs ` φpLqPrAs. Consequently, (2.68) defines a measure µ on pΩ, Fx,L q. Consider the increasing sequence tAn : n ě 1u of Fx,L -measurable sets defined by " * 1 1 1 An :“ ω P Ω : ErPx,ω rD “ 8s|Fx,L s ą P0 rD “ 8s ` φpLq ` n and define A :“ ď An . ně1 Observe that for each n ě 1 we have that 0 ď µpAn q “ pP0 rD “ 8s ` φpLqqPrAn s ´ ErErPx,ω rD1 “ 8s|Fx,L s1An s ď ´ n1 PrAn s. 82 Therefore, one has that for each n ě 1, PrAn s “ 0 and consequently PrAs “ 0. Observing that A “ tω P Ω : ErPx,ω rD1 “ 8s|Fx,L s ą P0 rD1 “ 8s ` φpLqu, we see that ErPx,ω rD1 “ 8s|Fx,L s ´ P0 rD1 “ 8s ď φpLq. (2.69) One can prove that ´φpLq ď ErPx,ω rD1 “ 8s|Fx,L s ´ P0 rD1 “ 8s following the same argument used to show (2.69), but changing the event tD1 “ 8u by tD1 ă 8u. The second lemma that will be needed to prove Proposition 2.5.3 is the following one. To state it define M :“ sup pXn ´ X0 q ¨ u, 0ďnďD1 D1 p0q :“ inftn ě 0 : Xn R Cp0, l, αqu, and for a P R Tal :“ inftn ě 0 : Xn ¨ l ě au T̄al :“ inftn ě 0 : Xn ¨ l ą au. and (2.70) Lemma 2.5.5. Let M ą 4d ` 1 and 2c ` 1 ď 1 . α (2.71) Assume that pP CqM,c |l is satisfied. Then, there exists c6 “ c6 pdq ą 0 such that a.s. one has that E0 rM2 , D1 ă 8|F0,L s ď c6 . 83 P´ almost surely. Proof. To simplify the proof, we will show that the second moment of M1 :“ sup pXn ´ X0 q ¨ l 0ďnďD1 is bounded from above. Note that E0 rM1 2 , D1 ă 8|F0,L s ď P0 rD1 ă 8 | F0,L s ` ř mě0 22pm`1q P0 r2m ď M1 ă 2m`1 , D1 ă 8 | F0,L s. (2.72) Therefore, it is enough to obtain an appropriate upper bound of the probability when m is large P0 r2m ď M1 ă 2m`1 , D1 ă 8 | F0,L s. Note that, P0 r2m ď M1 ă 2m`1 , D1 ă 8 | F0,L s ď P0 rT2lm ă D1 ă 8, T2lm`1 ˝ θT2m ą D1 p0q ˝ θT2m | F0,L s ď P0 rXT2lm R B ` B2m ,c2m ,l p0q, T2lm ă D1 ă 8 | F0,L s `P0 rXT2lm P B ` B2m ,c2m ,l p0q, T2lm`1 ˝ θT2m ą D1 p0q ˝ θT2m | F0,L s. (2.73) Using pP CqM,c |l, we get the following upper bound for the first term of the rightmost expression in (2.73), P0 rXTB2m ,c2m ,l p0q R B ` B2m ,2m ,l p0q, pXn q0ďnďTBi p0q Ă pH0,l qc |H0,l s ď 2´M m . (2.74) As for the second term in the rightmost expression in (2.73), it will be useful to introduce the set Fm :“ B ` B2m ,c2m ,l p0q. Now, by the strong Markov property we have the bound 84 P0 rXTB2m ,c2m ,l p0q P B ` B2m ,2m ,l p0q, T2lm`1 ˝ θT2lm ą D1 p0q ˝ θT2lm | F0,L s ř ď yPFm Py rT2lm`1 ą D1 p0q | F0,L s. (2.75) In order to estimate this last conditional probability, we obtain a lower bound for its complement as follows. To simplify the computations which follow, for each x P Zd we introduce the notation Bx :“ B2m´1 ,c2m´1 ,l pxq. Now, note that under the assumption (2.71) we have that ` ˘ c 2m ` 2m´1 ď cotpβq2m´1 , which implies that the boxes By and Bz , for all y P Fm and z P B ` By , are inside the cone Cp0, l, αq (see Figure 2.3). C(0, l, α) Bz By l B2m ,c2m ,l (0) Figure 2.3: The boxes By and Bz are inside of Cp0, l, αq. Therefore, fixing y P Fm , it follows that Py rT2lm`1 ă D1 p0q | F0,L s ě ř ` zPB ` By ErPy,ω rXTBy P B By , XTBy “ z, pXTBz P B ` Bz q ˝ θTBy s|F0,L s. 85 (2.76) To estimate the right-hand side of the above inequality, it will be convenient to introduce the set F̄m :“ BrYyPFm By s X tRpr2m´1 ` 2m , 8q ˆ Rd´1 qu, and the event GF̄m :“ tω P Ω : Pz,ω rXTBz P B ` Bz s ą 1 ´ 2´ M pm´1q 2 , for all z P F̄m u. Using the strong Markov property, we can now bound from below the right-hand side of inequality (2.76) by p1 ´ 2´ M pm´1q 2 ¯ ´ q Py rXTBy P B ` By |F0,L s ´ Py rpGF̄m qc |F0,L s . (2.77) In turn, by means of the polynomial condition and the fact that the boxes By and Bz are inside the cone Cp0, l, αq we see that (2.77) is greater than or equal to p1 ´ 2´ M pm´1q 2 ` ˘ q 1 ´ 2´M pm´1q ´ Py rpGF̄m qc |F0,L s . (2.78) Now, note that Py rpGF̄m qc |F0,L s ď ď |F̄m |2´ ř xPF̄m M pm´1q 2 2 M pm´1q 2 Px rXTBx R B ` Bx |F0,L s ď p4cqd´1 2mpd´1q 2´ M pm´1q 2 . (2.79) where in the first inequality we have used Chebyshev inequality, in the second one the assumption that pP CqM,c |l is satisfied and in the third one the bound |F̄2m | ď p4cqd´1 2mpd´1q . Consequently inserting the estimates (2.79) into (2.78) and combining this with inequality (2.76) we conclude that Py rT2lm`1 ď D1 p0q | F0,L s ě p1 ´ 2´ p1 ´ 2´ M pm´1q 2 M pm´1q 2 ´ p4cqd´1 2mpd´1q 2´ ě 1 ´ 3p4cqd´1 2mpd´1q 2´ 86 M pm´1q 2 M pm´1q 2 . qˆ q (2.80) Using the bound (2.80) in (2.75), together with the estimate |Fm | ď p2cqd´1 2mpd´1q , we see that P0 rXTB2m ,c2m ,l p0q P B ` B2m ,2m ,l p0q, T2lm`1 ˝ θT2lm ą D1 p0q ˝ θT2lm | F0,L s ď 3p4cq2pd´1q 22mpd´1q 2´ M pm´1q 2 . (2.81) Combining the estimates (2.81), (2.74), (2.73) with (2.72) we conclude that E0 rM1 2 , D1 ă 8|F0,L s ř M pm´1q ď 1 ` 4p4cq2pd´1q mě0 22pm`1q 22mpd´1q 2´ 2 ř ď 1 ` 4p4cq2pd´1q mě0 2´m ď c6 , where in the second to last inequality we have used the fact that M ą 4d ` 1 and c6 is a constant that does not depend on L. This completes the proof of the Lemma. 2.5.3 Proof of Proposition 2.5.3 To simplify the computations, we introduce the notation b “ bpLq :“ P0 pD1 ă 8q ` φpLq, b1 “ b1 pLq :“ P0 pD1 “ 8q ` φpLq and EPbQ :“ EEQ . Furthermore, it will be necessary to define for each j ě 0 and n ě L`j the events Dj,n :“ tε P W N : pεm , . . . , εm`L´1 q ‰ εpLq for all j ď m ď j ` n ´ L ` 1u. The following lemma, whose proof is presented in Appendix 2.7, will be useful in the proof of Proposition 2.5.3. Lemma 2.5.6. There exists a constant c7 such that for all n ě L2 one has that n QrD0,n s ď p1 ´ c7 L2 κL qr L2 s . 87 We now present the proof of Proposition 2.5.3, divided in several steps. For the sake of simplicity, we will write τ instead of τ pLq . Step 0. We first note that Ē0 rpXτ ¨ uq2 | F0,L s “ 8 k´1 ÿ ÿ Ē0 rpXSk1 `1 ¨ uq2 ´ pXSk1 ¨ uq2 , Sk ă 8, D1 ˝ θSk “ 8 | F0,L s. k“1k1 “0 (2.82) Throughout the subsequent steps of the proof we will estimate the right-hand side of (2.82). Step 1. Here we will prove the following estimate valid for all k ě 1 and 0 ď k 1 ă k. Ē0 rpXSk1 `1 ¨ uq2 ´ pXSk1 ¨ uq2 , Sk ă 8, D1 ˝ θSk “ 8 | F0,L s 1 ď b1 bk´k ´1 Ē0 rpXSk1 `1 ¨ uq2 ´ pXSk1 ¨ uq2 , Sk1 `1 ă 8 | F0,L s. (2.83) Furthermore, define the set * " |u|2 d . H :“ y P Z : y ¨ u ě L |u|1 L Then, for each 0 ď k 1 ă k, one has that Ē0 rpXSk1 `1 ¨ uq2 ´ pXSk1 ¨ uq2 , Sk ă 8, D1 ˝ θSk “ 8 | F0,L s ÿ “ EPbQ rEω,ε rpXSk1 `1 ¨ uq2 ´ pXSk1 ¨ uq2 , Sk “ n, ně1,xPH L XSk “ x, D1 ˝ θn “ 8 | F0,L s ÿ EPbQ rEω,ε rpXSk1 `1 ¨ uq2 ´ pXSk1 ¨ uq2 , Sk “ n, Xn “ xs “ ně1,xPH L Pϑx ω, ÿ “ θn ε rD 1 “ 8s | F0,L s ErĒ0,ω rpXSk1 `1 ¨ uq2 ´ pXSk1 ¨ uq2 , Sk ă 8, XSk “ xs xPH L Px,ω rD1 “ 8s | F0,L s, (2.84) 88 where here for each x P Zd , ϑx denotes the canonical space shift in Ω so that ϑx ωpyq “ ωpx ` yq, while for each n ě 0, θn denotes the canonical time shift in the space W so that pθn qm “ n`m , in the first equality we have used the fact that the value of XSk ¨ u ě XS1 ¨ u, in the second equality the Markov property and in the last equality we have used the independence of the coordinates of and the fact that the law of the random walk is the same under Px,ω and under EQ Pϑx ω,θn . Moreover, by the fact that the first factor inside the expectation of the right-most expression of (2.84) is Fx,L -measurable, the right-most expression in (2.84) is equal to ÿ ErĒ0,ω rpXSk1 `1 ¨ uq2 ´ pXSk1 ¨ uq2 , Sk ă 8, XSk “ xs xPH L ErPx,ω rD1 “ 8s | Fx,L s | F0,L s. (2.85) Applying next Lemma 2.5.4 to (2.85), we see that ř xPH L ErĒ0,ω rpXSk1 `1 ¨ uq2 ´ pXSk1 ¨ uq2 , Sk ă 8, XSk “ xs ˆErPx,ω rD1 “ 8s | Fx,L s | F0,L s ď b1 Ē0 rpXSk1 `1 ¨ uq2 ´ pXSk1 ¨ uq2 , Sk ă 8 | F0,L s. (2.86) Next, observe that for k 1 ă k one has that Ē0 rpXSk1 `1 ¨ uq2 ´ pXSk1 ¨ uq2 , Sk ă 8 | F0,L s “ Ē0 rpXSk1 `1 ¨ uq2 ´ pXSk1 ¨ uq2 , Rk´1 ă 8 | F0,L s “ ř xPH L ErĒ0,ω rpXSk1 `1 ¨ uq2 ´ pXSk1 ¨ uq2 , Sk´1 ă 8, XSk´1 “ x, D1 ˝ θSk´1 ă 8s | F0,L s “ ř xPH L ErĒ0,ω rpXSk1 `1 ¨ uq2 ´ pXSk1 ¨ uq2 , Sk´1 ă 8, XSk´1 “ xs Px,ω rD1 ă 8s | F0,L s “ ř xPH L ErĒ0,ω rpXSk1 `1 ¨ uq2 ´ pXSk1 ¨ uq2 , Sk´1 ă 8, XSk´1 “ xs ErPx,ω rD1 ă 8s | Fx,L s | F0,L s. (2.87) By Lemma 2.5.4, we have that ErPx,ω rD1 ă 8s | Fx,L s ď b “ P0 rD1 ă 8s ` φpLq. Using this inequality to estimate the last term in (2.87), we see that 89 Ē0 rpXSk1 `1 ¨ uq2 ´ pXSk1 ¨ uq2 , Sk ă 8 | F0,L s ď bĒ0 rpXSk1 `1 ¨ uq2 ´ pXSk1 ¨ uq2 , Sk´1 ă 8 | F0,L s. By induction on k we get that Ē0 rpXSk1 `1 ¨ uq2 ´ pXSk1 ¨ uq2 , Sk ă 8 | F0,L s 1 ď bk´k ´1 Ē0 rpXSk1 `1 ¨ uq2 ´ pXSk1 ¨ uq2 , Sk1 `1 ă 8 | F0,L s. (2.88) Combining (2.88) with (2.86) we obtain (2.83). Step 2. For k ě 1 we define Mk :“ sup Xn ¨ u. (2.89) 0ďnďRk Define also the sets parametrized by k and n ě 0 ´ ¯ ) ! An,k :“ ε P W N : εtpnq , εtpnq `1 , . . . , εtpnq `L´1 “ εpLq (2.90) ! ´ ) ¯ N pLq :“ ε P W : εtpjq , εtpjq `1 , . . . , εtpjq `L´1 ‰ ε for all 0 ď j ď n ´ 1 , (2.91) k k k and Bn,k k k k where we define the sequence of stopping times [c.f. (2.70)] parameterized by k and recursively on n ě 0 by p0q l tk :“ T̄M k and the successive times where a record value of the projection of the random walk on l is achieved by pn`1q tk :“ T̄Xl pnq ¨u . t k In this step we will show that for all k ě 0 one has that 90 Ē0 rpXSk`1 ¨ uq2 ´ pXSk ¨ uq2 , Sk`1 ă 8|F0,L s ` ď ř8 řL2 ´1 n“0 pnq Ē0 rpXSk`1 ¨ uq2 ´ pXSk ¨ uq2 , tk ă 8, An,k | F0,L s pnq n“L2 Ē0 rpXSk`1 ¨ uq2 ´ pXSk ¨ uq2 , tk ă 8, Bn,k , An,k | F0,L s, (2.92) To prove (2.92), we have to introduce some further notations. Now, note that on the event An,k X Bn,k one has that pnq Sk`1 “ tk ` L. Thus, as a consequence of the definition of Sk`1 , one has that P̄0 -a.s. ď tSk`1 ă 8u Ă pnq ttk ă 8, Bn,k , An,k u. (2.93) ně0 Display (2.92) now follows directly from (2.93). Step 3. Here we will derive an upper bound for the two sums appearing in the right-hand side in (2.92). In fact, we will prove that there is a constant c8 such that for all k ě 1 one has that řL2 ´1 pnq Ē0 rpXSk`1 ¨ uq2 ´ pXSk ¨ uq2 , tk ă 8, An,k | F0,L s ` ˘ ď c8 κL L4 bk´1 ` L2 Ē0 rXSk ¨ u, Sk ă 8|F0,L s n“0 (2.94) and ř8 n“L2 pnq Ē0 rpXSk`1 ¨ uq2 ´ pXSk ¨ uq2 , tk ă 8, Bn,k , An.k | F0,L s ` ř n L L r L2 s pn ` Lq2 bk´1 ď c8 8 κ p1 ´ c κ q 2 7 n“L ˘ `pn ` LqĒ0 rXSk ¨ u, Sk ă 8|F0,L s . Note that for all n ě 0 one has that Xtpn`1q ¨ u ď Xtpnq ¨ u ` |u|8 , k k 91 (2.95) and hence by induction on n we get that Xtpnq ¨ u ď Mk ` pn ` 1q|u|8 . k Therefore, if we set L1 :“ L|u| ` |u|8 ď c9 L, |u|1 (2.96) pnq where c9 is a constant depending on l and d, we can see that P0 -a.s on the event ttk ă 8, An,k u one has that XSk`1 ¨ u ď Nk,n :“ Mk ` n|u|8 ` L1 . (2.97) Therefore, for all 0 ď n ď L2 ´ 1 one has that pnq Ē0 rpXSk`1 ¨ uq2 ´ pXSk ¨ uq2 , tk ă 8, An,k | F0,L s pnq 2 ď Ē0 rNk,n ´ pXSk ¨ uq2 , tk ă 8, An,k | F0,L s ř ř 2 2 “ 8 j“0 xPZd EPbQ rEω,ε rNk,n ´ pXSk ¨ uq , pnq tk “ j, Xj “ xs1tpεj ,...,εj`L´1 q“εpLq u | F0,L s 2 ´ pXSk ¨ uq2 , Rk ă 8 | F0,L s, ď κL Ē0 rNk,n (2.98) where in the equality we have applied the Markov property and in the second inequality pnq the fact that Q is a product measure and that Rk ď tk . Similarly for all n ě L2 one has that pnq Ē0 rpXSk`1 ¨ uq2 ´ pXSk ¨ uq2 , tk ă 8, Bn,k , An,k | F0,L s pnq 2 ď Ē0 rNk,n ´ pXSk ¨ uq2 , tk ă 8, Bn,k , An,k | F0,L s ř ř8 ř 2 2 ď 8 j“0 j 1 “j`n yPZd EPbQ rEω,ε rNk,n ´ pXSk ¨ uq , p0q pnq Xtp0q “ y, tk “ jsPθy ω,θj ε rDj,n , tk “ j 1 s1tpεj1 ,...,εj1 `L´1 q“εpLq u s | F0,L s k 2 ď κL QrD0,n sĒ0 rNk,n ´ pXSk ¨ uq2 , Rk ă 8 | F0,L s n 2 ď κL p1 ´ c7 L2 κL qr L s Ē0 rNk,n ´ pXSk ¨ uq2 , Rk ă 8 | F0,L s, 92 (2.99) where in the second inequality we have used the Markov property, in the third one the p0q fact that Rk ď tk and in the last one Lemma 2.5.6. Now, by displays (2.98) and (2.99), to finish the proof of inequalities (2.94) and (2.95) it is enough to prove that there is a constant c10 such that 2 ´ pXSk ¨ uq2 , Rk ă 8 | F0,L s Ē0 rNk,n ` ˘ ď c10 pn ` Lq2 bk´1 ` pn ` LqĒ0 rXSk ¨ u, Sk ă 8|F0,L s , (2.100) using the fact that n ď L2 ´ 1 in the left-hand side of inequality (2.94). To prove (2.100), the following identity will be useful 2 Nk,n ´ pXSk ¨ uq2 “ pMk ´ XSk ¨ uq2 `2pn|u|8 ` L1 qpMk ´ XSk ¨ uq ` 2pn|u|8 ` L1 qXSk ¨ u `2pMk ´ XSk ¨ uqXSk ¨ u ` pn|u|8 ` L1 q2 . (2.101) We will now insert this decomposition in the left-hand side of (2.100) and bound the corresponding expectations of each term. Let us begin with the expectation of the last term. Note that by an argument similar to the one developed in Step 1 we have that Ē0 rpn|u|8 ` L1 q2 , Rk ă 8|F0,L s ď c11 pn ` Lq2 bk , (2.102) for some constant c11 . Similarly, the expectation of the first term of the right-hand side of display (2.101) can be bounded using Lemma 2.5.5, so that Ē0 rpMk ´ XSk ¨ uq2 , Rk ă 8 | F0,L s ÿ “ ErP̄0,ω rSk ă 8, XSk “ xsEx rM2 , D1 ă 8 | Fx,L s | F0,L s xPH L ď c6 bk´1 . (2.103) Again, for the expectation of the second term of the right-hand side of display (2.101), we have that Ē0 r2pn|u|8 ` L1 qpMk ´ XSk ¨ uq, Rk ă 8 | F0,L s ď c12 bk´1 pn ` Lq, 93 (2.104) for some suitable positive constant c12 . For the expectation of fourth term of the righthand side of (2.101), we see by Lemma 2.5.5 that Ē0 r2pMk ´ XSk ¨ uqXSk ¨ u, Rk ă 8 | F0,L s ? ď 2 c6 Ē0 rXSk ¨ u, Sk ă 8 | F0,L s. (2.105) Finally, for the expectation of third term of the right-hand side of (2.101) we have that Ē0 r2pn|u|8 ` L1 qXSk ¨ u, Rk ă 8 | F0,L s ď c12 bpn ` LqĒ0 rXSk ¨ u, Sk ă 8 | F0,L s. (2.106) Using the bounds (2.106), (2.105), (2.104), (2.103) and (2.102) we obtain inequality (2.100). Step 4. Here we will derive for all k ě 1 the inequality Ē0 rXSk ¨ u, Sk ă 8|F0,L s ´ řL2 ´1 ř pnq k´k1 ´1 ď k´1 n“0 Ē0 rNk1 ,n ´ XSk1 ¨ u, tk1 ă 8, An,k1 | F0,L s` k1 “0 b ¯ ř8 pnq n“L2 Ē0 rNk1 ,n ´ XSk1 ¨ u, tk1 ă 8, Bn,k1 , An,k1 | F0,L s . (2.107) Note that Ē0 rXSk ¨ u, Sk ă 8 | F0,L s “ řk´1 k1 “0 Ē0 rpXSk1 `1 ´ XSk1 q ¨ u, Sk ă 8 | F0,L s. (2.108) By an argument similar to the one used in Step 1 we see that for k 1 ă k one has that Ē0 rpXSk1 `1 ´ XSk1 q ¨ u, Sk ă 8 | F0,L s 1 ď bk´k ´1 Ē0 rpXSk1 `1 ´ XSk1 q ¨ u, Sk1 `1 ă 8 | F0,L s. (2.109) Now, we can use inclusion (2.93) in order to get that Ē0 rpXSk1 `1 ´ XSk1 q ¨ u, Sk1 `1 ă 8 | F0,L s ď ` řL2 ´1 n“0 ř8 n“L2 pnq Ē0 rpXSk1 `1 ´ XSk1 q ¨ u, tk1 ă 8, Bn,k1 , An,k1 | F0,L s pnq Ē0 rpXSk1 `1 ´ XSk1 q ¨ u, tk1 ă 8, Bn,k1 , An,k1 | F0,L s, 94 (2.110) where the events An,k1 and Bn,k1 are defined in (2.90) and (2.91). Using the fact that on pnq the event ttk1 ă 8, Bn,k1 , An,k1 u one has that P0 -a.s. pXSk1 `1 ´ XSk1 q ¨ u ď Nk1 ,n ´ XSk1 ¨ u, we see that the right-hand side of (2.110) is bounded by the right-hand side of (2.107), which is what we want to prove. Step 5. Here we will obtain an upper bound for the terms in the first summation in pnq (2.110). Indeed, note that on Rk1 ď tk1 , by an argument similar to the one used to derive inequality (2.98), we have that for all 0 ď n ď L2 and 0 ď k 1 ď k ´ 1 pnq Ē0 rNk1 ,n ´ XSk1 ¨ u, tk1 ă 8, An,k1 | F0,L s ď κL Ē0 rNk1 ,n ´ XSk1 ¨ u, Rk1 ă 8 | F0,L s. (2.111) Step 6. Here we will obtain an upper bound for the terms in the second summation in (2.110), showing that for all n ě L2 and 0 ď k 1 ď k ´ 1, pnq E0 rNk1 ,n ´ XSk1 ¨ u, tk1 ă 8, Bn,k1 , An,k1 | F0,L s ` ˘r n s ď κL 1 ´ c7 L2 κL L Ē0 rNk1 ,n ´ XSk1 ¨ u, Rk1 ă 8 | F0,L s. (2.112) Now note that pnq E0 rNk1 ,n ´ XSk1 ¨ u, tk1 ă 8, Bn,k1 , An,k1 | F0,L s ř ř ř ď 8 j“0 j 1 ěj`n yPZd EPbQ rEω,ε rNk1 ,n ´ XSk1 ¨ u, p0q pnq Xtp0q “ y, tk1 “ jsPθy ω,θj ε rDj,n , tk1 “ j 1 s1tpεj1 ,...,εj1 `L´1 q“εpLq u s | F0,L s k1 p0q “ κL QrD0,n sErĒ0,ω rNk1 ,n ´ XSk1 ¨ u, tk1 ă 8s | F0,L s. (2.113) Using Lemma 2.5.6 to estimate QrD0,n s we conclude the proof of inequality (2.112). Step 7. Here we will show that there exist constant c13 and c14 such that 2 ´1 Lÿ 1 pnq Ē0 rNk1 ,n ´ XSk1 ¨ u, tk1 ă 8, An,k1 | F0,L s ď c13 κL L4 bk ´1 n“0 95 (2.114) and 8 ÿ 1 pnq Ē0 rNk1 ,n ´ XSk1 ¨ u, tk1 ă 8, An,k1 , Bn,k1 | F0,L s ď 4c14 κ´L bk ´1 . (2.115) n“L2 Let us first note that by an argument similar to the one used to derive the bound in Step 1 (through Lemmas 2.5.4 and 2.5.5), we have that 1 Ē0 rNk1 ,n ´ XSk1 ¨ u, Rk1 ă 8s ď pn|u|8 ` L1 ` c15 qbk ´1 , where c15 :“ (2.116) ? c6 . Let us now prove (2.114). Indeed, note that by Step 5 and (2.116) we then have that řL2 ´1 pnq Ē0 rNk1 ,n ´ XSk1 ¨ u, tk1 ă 8, An,k1 | F0,L s ř 2 ´1 Ē0 rNk1 ,n ´ XSk1 ¨ u, Rk1 ă 8 | F0,L s ď κL Ln“0 n“0 1 ď c13 L4 κL bk ´1 , (2.117) for some suitable constant c13 . Let us now prove (2.115). First note that ř8 n“L2 ď pnq Ē0 rNk1 ,n ´ XSk1 ¨ u, tk1 ă 8, An,k1 , Bn,k1 | F0,L s n κL p1 ´ c7 L2 κL qr L s Ē0 rNk1 ,n ´ XSk1 ¨ u, Rk1 ă 8 | F0,L s ř n 1 L 2 L rLs ď bk ´1 8 pn|u|8 ` L1 ` c15 q n“L2 κ p1 ´ c7 L κ q ř n 1 L 2 L r L2 s ď c16 bk ´1 8 . n“L2 nκ p1 ´ c33 L κ q ř8 n“L2 (2.118) for some constant c16 , where in the first inequality we have used Step 6 and in the second we have used inequality (2.116). Finally notice that using the fact that for n ě L2 one “ ‰ has that n ď 2L2 Ln2 , we get that ř8 n“L2 “n‰ ř n n 2 L r L2 s nκL p1 ´ c7 L2 κL qr L2 s ď 2κL L2 8 n“L2 L2 p1 ´ c7 L κ q ř 2 2 L m ´L “ 2L4 κL 8 . m“1 mp1 ´ c7 L κ q ď pc7 q2 κ Using this estimate in (2.118) we obtain (2.115). 96 Step 8. Here we finish the proof of Proposition 2.5.3 combining the previous steps we have already developed. Combining inequality (2.107) proved in Step 4 with inequalities (2.114) and (2.115) proved in Step 7, we see that there is a constant c17 such that Ē0 rXSk ¨ u, Sk ă 8 | F0,L s ď c17 kbk´2 κ´L . (2.119) Thus, by inequality (2.94) proved in Step 3, we have that 2 ´1 Lÿ pnq Ē0 rpXSk`1 ¨ uq2 ´ pXSk ¨ uq2 , tk ă 8, An,k | F0,L s ď c18 L4 kbk´2 . (2.120) n“0 for certain positive constant c18 On the other hand, combining inequality (2.95) proved in Step 3 with (2.119), we see that there exists a constant c19 such that pnq ř8 n“L2 Ē0 rpXSk`1 ¨ uq2 ´ pXSk ¨ uq2 , tk ă 8, Bn,k , An,k | F0,L s ` ř n L 2 L r L2 s ď c19 8 pn ` Lq2 bk´1 n“L2 κ p1 ´ c7 L κ q ˘ `pn ` Lqkbk´2 κ´L . (2.121) Now, note that for some constant c20 one has that n ř8 n“L2 pn ř8 ` Lq2 p1 ´ c7 L2 κL qr L2 s ď c20 κ´3L n“L2 pn and n ` Lqp1 ´ c7 L2 κL qr L2 s ď c20 κ´2L . (2.122) (2.123) Substituting (2.122) and (2.123) into (2.121) we see that ř8 n“L2 pnq Ē0 rpXSk`1 ¨ uq2 ´ pXSk ¨ uq2 , tk ă 8, Bn,k , An,k | F0,L s ď c21 κ´2L bk´2 k, (2.124) for some suitable positive constant c21 . Substituting (2.121) and (2.124) into inequality (2.92) of Step 2, we then conclude that there is a constant c22 such that Ē0 rpXSk`1 ¨ uq2 ´ pXSk ¨ uq2 , Sk`1 ă 8|F0,L s ď c22 κ´2L bk´2 k. (2.125) Substituting (2.125) into (2.83) of Step 1, we get that Ē0 rpXSk1 `1 ¨ uq2 ´ pXSk1 ¨ uq2 , Sk ă 8, D1 ˝ θSk “ 8 | F0,L s ď b1 bk`1 k 1 . 97 (2.126) From the fact that ř8 řk´1 k1 “0 k“1 bk`1 k 1 ă 8 together with (2.126 and (2.82) of Step 0, we conclude that Ē0 rpXτ ¨ uq2 |F0,L s ď c23 κ´2L , for some constant c23 ą 0, which proves the proposition. 2.6 Proof of Theorem 2.1.1 In this section we will prove Theorem 2.1.1 using Proposition 2.5.3 proved in Section 2.5. First in Subsection 2.6.1, we will define an approximate sequence of regeneration times. In Subsection 2.6.2, we will show through this approximate regeneration time sequence, that there exists an approximate asymptotic direction. In Subsection 2.6.3, we will use the approximate asymptotic direction to prove Theorem 2.1.1. 2.6.1 Approximate regeneration time sequence pLq As in [CZ01], we define approximate regeneration by the recursively by τ1 :“ τ [c.f.(2.62)] and for i ě 2 pLq τi pLq :“ τ1 pLq We will drop the dependence in L on τ1 pLq ˝ θτ pLq ` τi´1 . i´1 when it is convenient for us, using the notation pLq τi instead τi . Let us define σ-algebras corresponding to the information of the random walk and the ε process up to the first regeneration time and of the environment ω at a distance of order L to the left of the position of the random walk at this regeneration time as pLq H1 :“ σpτ1 , X0 , ε0 , . . . , ετ pLq ´1 , Xτ pLq , 1 1 tωpy, ¨q : y ¨ u ă u ¨ Xτ pLq ´ L|u|{|u|1 uq. 1 Similarly define for k ě 2 98 pLq pLq Hk :“ σpτ1 , . . . , τk , X0 , ε0 , . . . , ετ pLq ´1 , Xτ pLq , k k tωpy, ¨q : y ¨ u ă u ¨ Xτ pLq ´ L|u|{|u|1 uq. (2.127) k Let us now recall Lemma 2.3 of [CZ01], stated here under the condition P0 rD1 “ 8s ą 0 [c.f. (2.36)] instead of Kalikow’s condition. Lemma 2.6.1. Let l P S˚ d´1 , α ą 0 and φ be such that limrÑ8 φprq “ 0. Consider a random walk in a random environment satisfying the cone-mixing assumption with respect to α, l and φ and uniformly elliptic with respect to l. Assume that L is such that φpLq ă P0 rD1 “ 8s. Then, P´a.s. one has that ˇ ˇP̄0 rtXτ k `¨ ˇ ´ Xτk u P A | Hk s ´ P̄0 rtX¨ u P A|D1 “ 8sˇ ď φ1 pLq, for all measurable sets A Ă pZd qN , where φ1 pLq :“ pP0 rD1 2φpLq . “ 8s ´ φpLqq Proof. For k “ 1, the argument given in page 890 of ([CZ01]) still works without any change. With the purpose of showing that the result continues being true under the weaker assumptions here, we complete the induction argument in the case k “ 2. To this end, we consider a positive H2 ´ measurable function h of the form h “ h1 ¨ ph2 q ˝ θτ1 (¨ denotes usual function multiplication), such that h1 , is H1 ´ measurable and h2 is H1 1 measurable, where the σ´ algebra H1 1 is defined as : pLq H1 1 : “ σpτ1 , X0 , ε0 , . . . , ετ pLq ´1 , Xτ pLq , 1 1 tωpy, ¨q : u ¨ y ď u ¨ Xτ pLq ´ L 1 99 |u| , y P CpX0 , l, αquq. |u|1 We let A be a measurable set of the path space, for short we will write 1A :“ 1tpXn ´X0 qně0 PAu . By the strong Markov property and using that τ1 ă 8 within an event of full P0 probability, we get: Ē0 rh1A ˝ θτ2 s ď ÿ E0 rh1A ˝ θτ2 s ně1 Ē0 rh1A ˝ θτ2 1K“n ˝ θτ1 , τ1 ă 8s ÿ Ē0 rh1A ˝ θτ2 , τ1 “ ts tě1 ÿ Ē0 rh1 ¨ ph2 ˝ θτ1 q1A ˝ θτ2 , St ă 8, D1 ˝ θSt “ 8s. (2.128) tě1 Now, notice that for given t P N, m P N, x P Zd , we can find a random variable h1,t,m,x |u| u, tXi uiăm q such that it coincides measurable with respect to σptωpy, ¨q : y ¨ u ă x ¨ u ´ L |u| 1 with h1 on the event tτ1 “ St “ m, XSt “ xu, therefore (2.128) equals ÿ Ē0 rh1,t,m,x ph2 ˝ θτ1 q1A ˝ θτ2 1St “m,D1 ˝θm “8,Xm “x s tě1,mě1,xPZd ÿ Ē0 rh1 1St “m,Xm “xD1 ˝θm “8 1A ˝ θτ2 h2 ˝ θτ1 s tě1,mě1,xPZd ÿ EPbQ rE0,ω,ε rh1,t,m,x 1St “m,Xm “x 1D1 ˝θm “8 1A ˝ θτ2 h2 ˝ θτ1 ss tě1,mě1,xPZd ÿ EPbQ rE0,ω,ε rh1,t,m,x 1St “m,Xm “x sEθx ω,θm ε r1D1 “8 1A ˝ θτ1 h2 ss. (2.129) tě1,mě1,xPZd We now work out the following expression Eθx ω,θm ε r1D1 “8 1A ˝ θτ1 h2 s ÿ Eθx ω,θm ε r1D1 “8 1A ˝ θτ1 h2 , Sn “ j, XSn “ z, D1 ˝ θj “ 8s. (2.130) zPCpx,l,αq ně1,jěm`1 Observe that, as in the case of h1 , for fixed x and m, we consider the probability measure Pθx ω,θm ε . Then we can find a measurable function h2,j,n,z with respect to σptωpy, ¨q : |u| , y P Cpx, l, αqu, tXi uiăj q , which coincides with h2 on the event y ¨ u ď z ¨ u ´ L |u| 1 tτ1 “ Sn “ j, XSn “ z, D1 “ 8u, furthermore note that D1 “ 8 depends up to pj ´ 1q coordinate in ε (recall that tD1 “ 8u P H1 ), hence we can apply the Markov property to get that the last expression in (2.130) is equal to: ÿ Eθx ω,θm ε rh2,j,n,z , 1Sn “j,XSn “z,D1 “8 sPθz ω,θj ε rA X tD1 “ 8us. zPCpx,l,αq ně1,jě1 100 (2.131) Using (2.131), it follows that (2.128) is equal to: ÿ ÿ tě1,mě1,xPZd zPCpx,l,αq ně1,jěm`1 EPbQ rE0,ω,ε rh1,t,m,x 1St “m,Xm “x s¨ Eθx ω,θm ε rh2,j,n,z , 1Sn “j,XSn “z,D1 “8 sPθz ω,θj ε rA X tD1 “ 8uss Following [CZ01], we can write down the expression above as ÿ ÿ EPbQ rrE0,ω,ε rh1,t,m,x 1St “m,Xm “x ss¨ tě1,mě1,xPZd zPCpx,l,αq ně1,jěm`1 Eθx ω,θm ε rh2,j,n,z 1Sn “j,XSn “z,D1 “8 ssP̄0 rA X tD1 “ 8us ` ρpAq, where ÿ ÿ tě1,mě1,xPZd zPCpx,l,αq ně1,jěm`1 ρpAq :“ CovPbQ rft,m,x,j,n,z , gj,z s, with: ft,m,x,j,n,z :“ E0,ω,ε rh1,t,m,x 1St “m,Xm “x sEθx ω,θm ε rh2,j,n,z , 1Sn “j,XSn “z,D1 “8 s and gj,z :“ Pθz ω,θj ε rA X tD1 “ 8us. On the other hand, since assumption pCM qφ,α |l, the estimate ÿ ρpAq ď φpLq ÿ EPbQ rE0,ω,ε rh1,t,m,x 1St “m,Xm “x s¨ tě1,mě1,xPZd zPCpx,l,αq ně1,jěm`1 Eθx ω,θm ε rh2,j,n,z 1Sn “j,XSn “z,D1 “8 ss holds for all measurable set A in the path space, in particular applying this for A “ Zd turns out the estimate: ÿ ÿ tě1,mě1,xPZd zPCpx,l,αq ně1,jěm`1 EPbQ rE0,ω,ε rh1,t,m,x 1St “m,Xm “x s¨ Eθx ω,θm ε rh2,j,n,z 1Sn “j,XSn “z,D1 “8 ss ď pP0 rD1 “ 8s ´ φpLqq´1 Ē0 rhs. From now on, we can follow the same sort of argument as in ([CZ01]), in order to conclude that k P̄0 rtXτ2 `n ´ Xτ2 u P ¨ | H2 s ´ P̄0 rtXn u P ¨ | D1 “ 8s kvar ď φ1 plq. 101 Therefore the second step induction is complete. 2.6.2 Approximate asymptotic direction We will show that a random satisfying the cone mixing, uniform ellipticity assumption and the non-effective polynomial condition with high enough degree has an approximate asymptotic direction. The exact statement is given below. It will also be shown that the right order in which the random variable Xτ1 grows as a function of L is κ´L . Proposition 2.6.2. Let l P S˚ d´1 , φ be such that limrÑ8 φprq “, c ą 0, M ą 6d and 1 0 ă α ă mint 91 , 2c`1 u. Consider a random walk in a random environment satisfying the cone mixing condition with respect to α, l and φ and the uniform ellipticity condition with respect to l. Assume that pP CqM,c |l is satisfied. Then, there exists a sequence ηL such that limLÑ8 ηL “ 0 and P̄0 -a.s. ˇ L ˇ ˇ κ X τn ˇ lim sup ˇˇ ´ λL ˇˇ ă ηL , n nÑ8 (2.132) λL :“ Ē0 rκL Xτ1 | D1 “ 8s. (2.133) |λL |2 ě c270 κ´L , (2.134) where for all L ě 1, Furthermore, for some constant c270 . We first prove inequality (2.132) of Proposition 2.6.2. We will follow the argument presented for the proof of Lemma 3.3 of [CZ01]. For each integer i ě 1 define the sequence X i :“ κL pXτi ´ Xτi´1 q, with the convention τ0 “ 0. Using Lemma 2.6.1 and Lemma 3.2 of [CZ01], we can enlarge the probability space where the sequence tXi : i ě 1u so that there we have the following properties: 102 ri , ∆i q : i ě 2u of random vectors with values in (1) There exist an i.i.d. sequence tpX r2 has the same distribution as X 1 under the measure pκL Zd , t0, 1uq, such that X P̄0 r¨|D1 “ 8s while ∆2 has a Bernoulli distribution on t0, 1u with P̄0 r∆i “ 1s “ φ1 pLq. (2) There exists a sequence tZi : i ě 2u of random variables such that for all i ě 2 one has that ri ` ∆i Zi . X i “ p1 ´ ∆i qX (2.135) Furthermore, for each i ě 2, ∆i is independent of Zi and of Gi :“ σtX j : j ď i ´ 1u. ri : We will call P the common probability distribution of the sequences tX i : i ě 2u, tX i ě 2u, tZi : i ě 2u and t∆i : i ě 2u, and E the corresponding expectation. From (2.135) note that n n n n ÿ 1ÿ X1 1 ÿ r 1ÿ ri ` 1 Xi “ ` Xi ´ ∆i X ∆i Zi . n i“1 n n i“2 n i“2 n i“1 (2.136) Let us now examine the behavior as n Ñ 8 of each of the four terms in the left-hand side of (2.136). Clearly, the first term tends to 0 as n Ñ 8. For the second term, note that on the event tD1 “ 8u, one has that | X 1 |22 ď c24 pX 1 ¨ lq2 for some constant c24 . r2 has the same distribution as X 1 Therefore, by Proposition 2.5.3, and the fact that X under P̄0 r¨|D1 “ 8s, we see that r2 |2 s “ Ē0 r|X 1 |2 |D1 “ 8s ď c24 Ē0 rpX 1 ¨ lq2 |D1 “ 8s ă c25 , Er|X 2 2 (2.137) for a suitable constant c25 . Hence, by the strong law of large numbers, we actually have that P -a.s. n 1ÿ r lim Xi “ λL . nÑ8 n i“2 103 (2.138) For the third term in the left-hand side of (2.136) we have by Cauchy-Schwartz inequality that ˇ ˇ ˜ ¸ 12 ˜ ¸ 12 n n n ˇ1 ÿ ˇ ÿ ÿ 1 1 ˇ r ˇˇ ď r i |2 ∆X |X ∆i . ˇ ˇ n i“2 i i ˇ n i“2 n i“2 (2.139) 2 Again by (2.137) and Proposition 2.5.3, we know that there is a constant c26 [c.f. (2.66)] such that P -a.s. n 1ÿ r 2 |Xi |2 “ Ē0 r|X 1 |22 |D1 “ 8s ď c26 . nÑ8 n i“2 lim As a result, from (2.139) we see that ˇ ˇ n ˇ ˇ1 ÿ a ˇ ri ˇˇ ď c26 φ1 pLq. ∆i X lim sup ˇ ˇ nÑ8 ˇ n i“2 (2.140) 2 pLq For the fourth term of the left-hand side of (2.136), we note setting Z i :“ ErZi | Gi s that Mnj :“ n ÿ ∆i pZi ´ Z i q ¨ ej i i“2 for n ě 2, j P t1, 2, . . . , nu is a martingale with mean zero with respect to the filtration tGi : i ě 1u. Thus, from the Burkholder-Gundy inequality [W91], we know that there is a constant c27 such that for all j P t1, 2, . . . , du «ˆ E ˙2 ff sup Mnj n « ď c27 E 8 ÿ |∆i pZi ´ Z i q|2 ff 2 i“2 . i2 (2.141) Now, since (2.135), note that for all i ě 2, |∆i Zi | ď |X̄i |. It follows that there exists a constant c28 such that Er|Zi |22 |Gi s ď 1 φ1 pLq E0 r|X 1 |22 , D1 “ 8|F0,L s ď 1 c28 , φ1 pLq (2.142) where we have used Proposition 2.5.3 and Lemma 2.5.4 in the second inequality. So that by (2.141) we see that the martingale tMnj : n ě 1u converges P ´a.s. to a random variable for any j P t1, 2, . . . , du. Thus, by Kronecker’s lemma applied to each component j P t1, 2, . . . , du, we conclude that P -a.s. 104 n 1ÿ ∆i pZi ´ Z i q “ 0. nÑ8 n i“2 lim (2.143) Now, note from (2.142) that there is a constant c29 such that 1 1 |Z i |2 ď Er|Zi |22 | Gi s 2 ď c29 φ1 pLq´ 2 . (2.144) Therefore, P -a.s. we have that ˇ ř ˇ 1 lim supnÑ8 ˇ n1 ni“2 ∆i Z i ˇ2 ď c29 φ1 pLq´ 2 lim supnÑ8 1 ď c29 φ1 pLq 2 . 1 n řn i“1 ∆i (2.145) Substituting (2.145), (2.140) and (2.138) into (2.136), we conclude the proof of inequality 1 (2.132) provided we set ηL “ c30 φ1 pLq 2 for some constant c30 . Let us now prove the inequality (2.134). By an argument similar to the one presented in [CZ01] to show that the random variable τ1 has a lower bound of order κ´L , we can ř show that Xτ1 ¨ l is bounded from below by the sum S :“ N i“1 Ui , where tUi : i ě 1u are i.i.d. random variables taking values on t1, 2, . . . , Lu with law P rUi “ ns “ κn for 1 ď n ď L, while N :“ minti ě 1 : Ui “ Lu. It is clear then that ErXτ1 ¨ ls ě ErN s “ c31 κ´L , for some constant c31 . 2.6.3 Proof of Theorem 2.1.1 It will be enough to prove that there is a constant c32 such that for all L ě 1 one has that ˇ ˇ ˇ Xn ˇ λ L ˇ ă c260 ηL . lim sup ˇˇ ´ (2.146) |Xn |2 |λL |2 ˇ2 λL nÑ8 Indeed, by compactness, we know that we can choose a sequence tLm , m ě 1u such that λLm “ v̂, mÑ8 |λLm |2 lim 105 (2.147) exists. On the other hand, by the inequality (2.134) of Proposition 2.5.3, we know that limmÑ8 ηLm λLm “ 0. Now note that by the triangle inequality and (2.146), for every m ě 1 one has that ˇ ˇ ˇ ˇ ˇ ˇ λLm ˇ ˇ Xn η L m lim sup ˇˇ ´ v̂ ˇˇ ď c32 ` ˇˇ ´ v̂ ˇˇ . |Xn |2 λLm |λLm |2 nÑ8 2 2 (2.148) Taking the limit m Ñ 8 in (2.148) using (2.147) we prove Theorem 2.1.1. Let us hence prove inequality (2.146). Choose a nondecreasing sequence tkn : n ě 1u, P - a.s. tending to `8 so that for all n ě 1 one has that τkn ď n ă τkn `1 . Notice that Xn “ |Xn |2 ˆ Xn ´ Xτkn |Xn |2 ˙ ˆ ` Xτkn kn kn |Xn |2 ˙ . (2.149) On the other hand, we assume for the time being, that for large enough L we have proved that lim sup nÑ8 |Xn ´ Xτkn |2 “ 0. kn (2.150) |Xn ´ Xτkn |2 “ 0. |Xn |2 (2.151) Note first that (2.150) implies that lim sup nÑ8 |l|2 Indeed, note that |Xn |2 ě Xn ¨ l ě Xτkn ¨ l ě kn L |l| , which in combination with (2.150) 1 implies (2.151). Also, from (2.150) and the fact that |Xτkn |2 |Xn ´ Xτkn |2 |Xτkn |2 |Xn ´ Xτkn |2 |Xn |2 ´ ď ď ` , kn kn kn kn kn (2.152) ˇ L ˇ ˇ κ |Xn |2 ˇ ˇ lim sup ˇ ´ |λL |2 ˇˇ ď ηL . kn nÑ8 (2.153) we see that 2 Combining (2.151) and (2.153) with (2.149) we get (2.146). Thus, it is enough to prove the claim in (2.150). To this end, note that 106 |Xpτkn `jq^τkn `1 ´ Xτkn |2 |Xn ´ Xτkn |2 ď sup kn kn jě0 (2.154) ` ˘ pkě1 :“ κL supjě0 |Xpτ `jq^pτ q ´ Xτ | We now consider the sequence X , a coupling k k k`1 kě1 decomposition as in the proof of Proposition 2.6.2 turns out; in a enlarged probability space P if necessary, the existence of two i.i.d. sequences pXk qkě1 , p∆k qkě1 and a sequence pYk qkě1 , such that P supports the following: p1 under P̄ r¨ | D1 “ 8s, and • For k ě 1, the common law of Xk is the same as X one has that ∆k is Bernoulli with values in the set t0, 1u independent of Gk and Pr∆k “ 1s “ φ1 pLq. • P- almost surely for k ě 1, we have the decomposition: pk “ p1 ´ ∆k qXk ` ∆k Yk X Furthermore, quite similar arguments as the ones given in the proof of Proposition 2.6.2 allow us to conclude that: n ÿ |Xj | x1 | | D1 “ 8s ă 8, Ñ Er|X n j“1 n ÿ ∆j pYj ´ Yrj q Ñ 0 and n j“1 n ÿ |∆j Yrj | 1 ď c240 φ1 pLq 2 . n j“1 (2.155) (2.156) where Yrj :“ ErYj | Gj s. Therefore, using the following inequality pk X Xk ∆k pYk ´ Yrk q ∆k Yrk “ ` ` , k k k k (2.157) Xk ÑkÑ8 0 k (2.158) implies that The proof is finished. 107 2.7 Appendix Proof of Lemma 2.5.6 Here we will prove Lemma 2.5.6. Let us first remark that it will be enough to show that there exists a constant c33 ą 0 such that for all L P |u|1 N QrD0,L2 s ď 1 ´ c33 L2 κL . (2.159) Indeed, using this inequality and the product structure of Q, for all n ě L2 one has that n QrD0,n s ď p1 ´ c7 L2 κL qr L2 s . In order to prove (2.159), for j “ L2 ´ L and i “ 0, 1, . . . , j consider the events Ai “ tε : pεi , . . . , εi`L´1 q “ εpLq u. Then, by the inclusion-exclusion principle we have that QrpD0,L2 qc s ě ÿ ÿ QrAj1 s ´ 0ďj1 ďj QrAj1 X Aj2 s. (2.160) 0ďj1 ăj2 ďj Now, note that ÿ QrAj1 X Aj2 s ď jκL`1 ` pj ´ 1qκL`2 ` . . . 0ďj1 ăj2 ďj . . . ` pj ´ L ` 1qκ2L ` pj ´ Lqκ2L ` . . . ` pj ´ pj ´ 1qqκ2L L ÿ 1 ´ κL`1 L ` L4 κ2L ď jκ κn ` κ2L pj ´ Lq2 ď L2 κL 1 ´ κ n“1 ď c34 L2 κL , (2.161) for some constant c34 . Since QrAi s “ κL for all 1 ď i ď j, we conclude from (2.160) and (2.161) that there is a constant c33 such that QrD0,L2 s “ 1 ´ QrpD0,L2 qc s ď 1 ´ c33 L2 κL . This finishes the proof. 108 Acknowledgments: I wish to thank for a private communication about Sznitman’s ballisticity conditions to Alexander Drewitz. 109 Bibliography [Al99] S. Alili. Asymptotic behaviour for random walks in random environments. J. Appl. Probab. 36, no. 2, 334-349, (1999). [BDR14] N. Berger, A. Drewitz and A.F. Ramı́rez. Effective Polynomial Ballisticity Conditions for Random Walk in Random Environment. Comm. Pure. Appl. Math. (2014). [Be08] N. Berger. Limiting velocity of high-dimensional random walk in random environment. Ann. 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