Sharp ellipticity conditions for ballistic behavior of random walks in random environment

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1 Bernoulli 222, 206, DOI: 0.350/4-BEJ683 arxiv:30.628v3 [math.pr] 5 Feb 206 Sharp ellipticity conditions for ballistic behavior of random walks in random environment ÉLODIE BOUCHET, ALEJANDRO F. RAMÍREZ2 and CHRISTOPHE SABOT Université de Lyon, Université Lyon, Institut Camille Jordan, CNRS UMR 5208, 43, Boulevard du novembre 98, Villeurbanne Cedex, France. bouchet@math.univ-lyon.fr; sabot@math.univ-lyon.fr 2 Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Macul, Santiago, Chile. aramirez@mat.puc.cl We sharpen ellipticity criteria for random walks in i.i.d. random environments introduced by Campos and Ramírez which ensure ballistic behavior. Furthermore, we construct new examples of random environments for which the walk satisfies the polynomial ballisticity criteria of Berger, Drewitz and Ramírez. As a corollary, we can exhibit a new range of values for the parameters of Dirichlet random environments in dimension d = 2 under which the corresponding random walk is ballistic. Keywords: Dirichlet distribution; max-flow min-cut theorem; random walk in random environment; reinforced random walks. Introduction We continue the study initiated in [3] sharpening the ellipticity criteria which ensure ballistic behavior of random walks in random environment. Furthermore, we apply our results to exhibit a new class of ballistic random walks in Dirichlet random environments in dimension d=2. For x R d, denote by x and x 2 its L and L 2 norm, respectively. Call U :={e Z d : e =}={e,...,e 2d } the canonical vectors with the convention that e d+i = e i for i d. We set P :={pe: pe 0, e U pe=}. An environment is an element ω:={ωx: x Z d } of the environment space Ω:=P Zd. We denote the components of ωx by ωx,e. The random walk in the environment ω starting from x is the Markov chain {X n : n 0} in Z d with law P x,ω defined by the condition P x,ω X 0 =x = and the transition This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 206, Vol. 22, No. 2, This reprint differs from the original in pagination and typographic detail c 206 ISI/BS

2 2 É. Bouchet, A.F. Ramírez and C. Sabot probabilities P x,ω X n+ =x+e X n =x=ωx,e for each x Z d and e U. Let P be a probability measure defined on the environment space Ω endowed with its Borel σ-algebra, such that {ωx: x Z d } is i.i.d. under P. We call P x,ω the quenched law of the random walk in random environment RWRE starting from x, and P x := Px,ω dpω the averaged or annealed law of the RWRE starting from x. The law P is said to be elliptic if for every x Z d and e U, Pωx,e>0=. We say that P is uniformly elliptic if there exists a constant γ >0 such that for every x Z d and e U, Pωx,e γ=. Given l S d we say that the RWRE is transient in direction l if with P 0 A l =, { } A l := lim X n l=. n Furthermore, it is ballistic in direction l if P 0 -a.s. lim inf n X n l n Given Λ Z d, we denote its outer boundary by >0. Λ:={x / Λ: x y = for some y Λ}. We denote any nearest neighbour path with n steps joining two points x,y Z d by x,x 2,...,x n, where x =x and x n =y... Polynomial condition, ellipticity condition In [], Berger, Drewitz and Ramírez introduced a polynomial ballisticity condition within the uniformly elliptic context, which was later extended to the elliptic case by Campos and Ramírez in [3]. This condition will be of interest for our results. It is effective, in the sense that it can a priori be verified explicitly for a given environment. To define it, we need for each L, L>0 and l S d to consider the box B l:=r L,L L, L d Z d, where R is a rotation of R d that verifies Re =l. For each subset A Z d we denote the first exit time from the set A as T A :=min{n 0: X n / A}.

3 Sharp ellipticity conditions for ballistic behavior 3 Define also the half space We now choose α>0 such that and let H l :={l R d : l l 0}. η α := max e H l Z d E ω0,e α < c 0 := d d/αlogη α 2.. Definition. Given M, we say that condition P M in direction l is satisfied also written as P M l if there exists L c 0 and L 70L 3 such that one has the following upper bound for the probability that the walk does not exit the box B l through its front side: P 0 X l<l TBl L M. This condition has proven to be useful in the uniformly elliptic case. Indeed, P M for M 5d+5 implies ballisticity see []. For non-uniformly elliptic environments in dimensions d 2, there exist elliptic random walks which are transient in a given direction but not ballistic in that direction see, e.g., Sabot Tournier [0], Bouchet [2]. In [3], Campos and Ramírez introduced ellipticity criteria on the law of the environment which ensure ballisticity if condition P M is satisfied for M 5d+5 In this article we will sharpen this ellipticity criteria. Remark. In definition.6 of [3], an incorrect value of the constant c 0 is given, different from the definition in.. Nevertheless, it is straightforward to check that the argument of Section 3. of [3] showing that P M implies T γl does not change. Letusfirstrecalltheellipticity conditionof[3].forallm,thepolynomialcondition P M implies the existence of an asymptotic direction see, e.g., Simenhaus []: there exists ˆv S d such that P 0 -a.s., We call ˆv the asymptotic direction. lim n X n X n 2 =ˆv. Definition 2. Let β >0. We say that the law of the environment satisfies the ellipticity condition E β if there exists an {αe: e U} 0, 2d such that κ{αe: e U}:=2 αe maxαe+α e>β.2 e U e

4 4 É. Bouchet, A.F. Ramírez and C. Sabot and for every e U E ω0,e <. αe.3 e Furthermore, when ˆv exists, we say that the ellipticity condition E β is satisfied towards the asymptotic direction if there exists an {αe: e U} satisfying.2 and.3 and such that there exists α >0 that satisfies αe=α for e Hˆv U while αe α for e U \Hˆv. Remark 2. In [3],.3 is replaced by E e e ω0,e αe <. Those two conditions are in fact equivalent. The direct implication is straightforward. And since e U {ω0,e /2d}, we get E This gives the reverse implication. e ω0,e αe e U2d αe E e e ω0,e αe <. Remark 3. Knowing the existence of ˆv does not mean that we know its value. In most cases, ˆv is found to be inaccessible. A notable exception is the result of Tournier [5] that gives the value of ˆv in the case of random walks in Dirichlet environments..2. Ballisticity results Our main results are a generalization of Theorems.2 and.3 of [3] where we remove the towards the asymptotic direction condition of Theorems.2 and.3 of [3]. Let τˆv be the first renewal time in the direction ˆv, its precise definition is recalled in the next section. We prove the following tail estimate on renewal times, which improves Proposition 5. of [3]. Theorem. Let l S d, β >0 and M 5d+5. Assume that P M l is satisfied and that E β holds cf..2,.3. Then lim suplogu logp τˆv 0 >u β. u TheconditionE β issharpinasensethatismadepreciseinremark4below.together with previous results of Sznitman, Zerner, Seppäläinen and Rassoul-Agha, cf. [8, 2, 4, 8], it implies the following. Theorem 2 Law of large numbers. Consider a random walk in an i.i.d. environment in dimensions d 2. Let l S d and M 5d+5. Assume that the random walk

5 Sharp ellipticity conditions for ballistic behavior 5 satisfies condition P M l and the ellipticity condition E. Then the random walk is ballistic in direction l and there is a v R d, v 0 such that X n lim n n =v, P 0-a.s. Theorem 3 Central limit theorems. Consider a random walk in an i.i.d. environment in dimensions d 2. Let l S d and M 5d+5. Assume that the random walk satisfies condition P M l. a Annealed central limit theorem. If E 2 is satisfied, then ε /2 X [ε n] [ε n]v converges in law under P 0 as ε 0 to a Brownian motion with non-degenerate covariance matrix. b Quenched central limit theorem. If E 76d is satisfied, then P-a.s. we have that ε /2 X [ε n] [ε n]v converges in law under P 0,ω as ε 0 to a Brownian motion with non-degenerate covariance matrix. Removing the towards the asymptotic direction is a real improvement: in Section.3.2, we will give some examples of environments in the class of Dirichlet environments that satisfy E β but not towards the asymptotic direction. For those environments, our new theorems allow to prove a LLN or CLT. Furthermore, our final goal would be to get a ballisticity condition that depends only locally on the environment i.e., a condition that depends only on the law of the environment at one point. Condition E β is local, whereas E β towards the asymptotic dimension is not: removing the towards the asymptotic direction is then a first step in this direction. Ideally, we would also need to get rid of condition P M l, that is not local either. This is a much more difficult problem, not solved even in the uniformly elliptic case. Remark 4. The condition of Theorem is sharp under the following assumption on the tail behavior of the environment at one site: there exists some β e e U, β e 0, and a positive constant C > such that for all e U C e U,e e t β e e Pω0,e t e, e U,e e C e U,e e t β e e.4 for all t e e U\{e}, 0 t e. Indeed, in this case we easily see that E β is satisfied if and only if β <2 e β e max e Uβ e +β e. On the other hand, if β 2 e β e max e U β e +β e then Eτˆv β =. Indeed, consider a direction e 0 which realizes the maximum in max e U β e +β e and set K ={0,e 0 }. We denote by + K the set of edges that exit the set K composed of the edges {0,e} e e0 and {e 0,e} e e0. For small

6 6 É. Bouchet, A.F. Ramírez and C. Sabot t>0, under the condition that ωx,y x t for all x,y + K we have P 0,ω T K n 2d t n. Hence, P 0 T K n 2d /n n Pωx,y x /n, x,y + K 2d /n n C n e β e βe 0 +β e 0 which implies that E 0 T β K =. Since T K is clearly a lower bound for the first renewal time it gives the result. Dirichlet environment cf. the next section is a typical example of environment that satisfies condition.4. Remark 5. Theorem. of [3] states that for i.i.d. environments in dimensions d 2 satisfying the ellipticity condition E 0, the polynomial condition P M l for l S d and M 5d + 5 is equivalent to Sznitman s condition T l see, e.g., [3] for the definition. We can therefore replace P M l by T l in the statements of Theorems 2 and New examples of random walks satisfying the polynomial condition In this article, we also introduce new examples of RWRE in environments which are not uniformly elliptic and which satisfy the polynomial condition P M for M 5d+5. In Section.3., we prove the polynomial condition for a subset of marginal nestling random walks, including a particular two-dimensional environment introduced by Campos and Ramírez in [3]. In Section.3.2, we prove the polynomial condition for a class of random walks in Dirichlet random environments which do not necessarily satisfy Kalikow s condition. In both cases, we present the case of environments for which our new Theorems 2 and 3 prove necessary to study the behaviour of the walks..3.. Example within the class of marginal nestling random walks Following Sznitman [2], we say that a law P on Ω is marginal nestling if the convex hull K o of the support of the law of d0,ω:= e Uω0,ee is such that 0 K o. We will prove in Section 4 that a certain subset of the marginal nestling laws satisfies the polynomial condition. Theorem 4. Consider an elliptic law P under which {ωx: x Z d } are i.i.d. Assume that there exists an r> such that ω0,e =rω0,e +d. Then the polynomial condition P M e is satisfied for some M 5d+5.

7 Sharp ellipticity conditions for ballistic behavior 7 Remark 6. This theorem is valid for all i.i.d. elliptic environments satisfying ω0,e = rω0,e +d, including uniformly elliptic environments. However, the environments are marginal nestling only in the non-uniformly elliptic case. The above result includes an example suggested in [3], by Campos and Ramírez, of an environment which satisfies the polynomial condition and for which the random walk is directionally transient but not ballistic. They showed that on this environment, E α is satisfied for α smaller but arbitrarily close to, and that the walk is transient but not ballistic in a given direction. The proof that this environment satisfies the polynomial condition was left for a future work. Let us define the environment introduced in [3]. Let ϕ be any random variable taking values on the interval 0,/4 and such that the expected value of ϕ /2 is infinite, while for every ε > 0, the expected value of ϕ /2 ε is finite. Let X be a Bernoulli randomvariableofparameter /2.We now define ω0,e =2ϕ, ω0, e =ϕ, ω0,e 2 = Xϕ+ X 4ϕ and ω0, e 2 =X 4ϕ+ Xϕ. For every ε>0, this environment satisfies E ε : traps can appear because the random walk can get caught on two edges of the type x,e 2,x+e 2, e 2. Furthermore, it is transient in direction e but not ballistic in that direction Examples within the class of Dirichlet random environments Random Walks in Dirichlet Environment RWDE are interesting because of the analytical simplifications they offer, and because of their link with reinforced random walks. Indeed, the annealed law of a RWDE corresponds to the law of a linearly directed-edge reinforced random walk [4, 7]. Given a family of positive weights β,...,β 2d, a random i.i.d. Dirichlet environment is a law on Ω constructed by choosing independently at each site x Z d the values of ωx,e i i [[,2d]] according to a Dirichlet law with parameters β,...,β 2d. That is, at each site we choose independently a law with density Γ 2d i= β i 2d i= Γβ i 2d i= x βi i dx dx 2d onthe simplex {x,...,x 2d ]0,] 2d, 2d i= x i =}.Here Γ denotesthe Gammafunction Γβ= 0 tβ e t dt, and dx dx 2d represents the image of the Lebesgue measure on R 2d by the application x,...,x 2d x,...,x 2d, x x 2d. Obviously, the law does not depend on the specific role of x 2d. Remark 7. Given a Dirichlet law of parameters β,...,β 2d, the ellipticity condition E β is satisfied if and only if κβ,...,β 2d =2 2d i= β i max i=,...,d β i+β i+d >β.

8 8 É. Bouchet, A.F. Ramírez and C. Sabot As stated in Remark 4, this ellipticity condition is optimal to get Theorem in the case of Dirichlet environments. Remark that for Dirichlet environments, for all β > 0, E β is much sharper that E β towards the asymptotic direction. Indeed, the result of Tournier [5] gives us the explicit value of ˆv in the case of Dirichlet laws: ˆv= 2d i= βiei 2d i= βiei. Without loss of generality, we can assume that β i β i+d for i d. This implies that e i ˆv 0 for i d. If we define β i :=min j d β j and β i+d :=min β i,β i+d for i d, we can see that E β is satisfied towards the asymptotic direction if and only if κ{ β i : i 2d}>β. In the case of RWDE, it has been proved that Kalikow s condition, and thus the T condition, is satisfied whenever d β i β i+d >.5 i= see Enriquez and Sabot in [5] and Tournier in [6]. The characterization of Kalikow s condition in terms of the parameters of a RWDE remains an open question. On the other hand, we believe that for RWDE condition T is satisfied if and only if max i d β i β i+d >0. Nevertheless, in this article we are able to prove the following result. Theorem 5. Let β,β 2,...,β d,β d+2,...,β 2d be fixed positive numbers. Then, there exists an ε 0, depending on these numbers such that if β +d is chosen so that β +d ε, the Random Walk in Dirichlet Environment with parameters β,...,β 2d satisfies condition P M e for M 5d+5. Theorem5givesas acorollarynew examplesofrwde which areballistic in dimension d=2 since they do not correspond to ranges of the parameters satisfying condition.5 of Tournier [6] and Sabot and Enriquez [5] see the following remark for the case d 3. Indeed, by Theorem 2, if 2 2d i= β i max i d β i+β i+d > and one of the parameters {β i : i d} is small enough, the walk is ballistic. Remark 8. In dimension d 3, in [2, 9], precise conditions on the existence of an invariant measure viewed from the particle absolutely continuous with respect to the law have been given; this allows to characterize completely the parameters for which there is ballisticity, but it fails to give information on the T condition and on the tails of renewal times. It also fails to give a CLT. Theorem 3 then gives us annealed CLTs for Dirichlet laws when the parameters β,...,β 2d satisfy 2 2d i= β i max i d β i +β i+d >2 along with condition.5 or the hypothesis of Theorem 5.

9 Sharp ellipticity conditions for ballistic behavior 9 Remark 9. For the Dirichlet laws in dimension d=2 with parameters β,...,β 4 satisfying 2 4 i= β i max i 2 β i +β i+d >, with one of the parameters {β i : i 4} small enough, but for which E is not satisfied toward the asymptotic direction, our Theorem 2 gives the ballisticity when the results of [3] would not have been enough. For the Dirichlet laws in dimension d 2 with parameters β,...,β 2d satisfying 2 2d i= β i max i d β i +β i+d >2,withcondition.5orthehypothesisofTheorem5, but for which E 2 is not satisfied toward the asymptotic direction, our Theorem 3 gives the annealed CLT when the results of [3] would not have been enough. This illustrates the relevance of having removed the toward the asymptotic direction hypothesis in Theorem. 2. First tools for the proofs In this section, we will introduce some tools that will prove necessary for the proof of Theorem. 2.. Regeneration times The proofs in [3] are based on finding bounds on the regeneration times. We thus begin by giving the definition and some results about the regeneration times with respect to a fixed direction l. In the following, we suppose that the walk is transient in direction l. We define {θ n : n } as the canonical time shift on Z d N. For l S d and u 0, we define the time and Set We define and recursively for k, T l u :=min{n 0: X n l u}. a>2 d 2. D l :=min{n 0: X n l<x 0 l}. S 0 :=0, M 0 :=X 0 l, S :=T l M 0+a, R :=D l θ S +S, M :=max{x n l: 0 n R }, S k+ :=T l M k +a, R k+ :=D l θ Sk+ +S k+, M k+ :=max{x n l: 0 n R k+ }.

10 0 É. Bouchet, A.F. Ramírez and C. Sabot The first regeneration time is then defined as τ :=min{k : S k <,R k = }. We can now define recursively in n the n+th regeneration time τ n+ as τ X + τ n X τ+ X τ. We will occasionally write τ l,τl 2,... to emphasize the dependence on the chosen direction. Remark 0. The condition 2. on a is only necessary to prove the non-degeneracy of the covariance matrix of part a of Theorem 3. It is a standard fact see, e.g., Sznitman and Zerner [4] to show that under the assumption of transience in direction l, the sequence τ,x τ+ τ 2 X τ,τ 2 τ,x τ2+ τ 3 X τ2,... is independent and except for its first term i.i.d. Its law is the same as the law of τ with respect to the conditional probability measure P 0 D l =. Those regeneration times are particularly useful to us because of the two following theorems. Theorem 6 Sznitman and Zerner [4], Zerner [8], Sznitman [2]. Consider a RWRE in an elliptic i.i.d. environment. Let l S d and assume that there is a neighbourhood V of l such that for every l V the random walk is transient in the direction l. Then there is a deterministic v such that P 0 -a.s. Furthermore, the following are satisfied. X n lim n n =v. a If E 0 τ <, the walk is ballistic and v 0. b If E 0 τ 2 <, ε /2 X[ε n] [ε n]v converges in law under P 0 to a Brownian motion with non-degenerate covariance matrix. Theorem 7 Rassoul-Agha and Seppäläinen [8]. Consider a RWRE in an elliptic i.i.d. environment. Take l S d and let τ be the corresponding regeneration time. Assume that E 0 τ p <, for some p>76d. Then P-a.s. we have that ε /2 X [ε n] [ε n]v converges in law under P 0,ω to a Brownian motion with non-degenerate covariance matrix.

11 Sharp ellipticity conditions for ballistic behavior 2.2. Atypical quenched exit estimate The proof of Theorem is based on an atypical quenched exit estimate proved in [3]. We will also need this result, and thus recall it in this section. Let us first introduce some notations. Without loss of generality, we can assume that e is contained in the open half-space defined by the asymptotic direction so that We define the hyperplane: ˆv e >0. H :={x R d : x e =0}. LetP :=Pˆv betheprojectionontheasymptoticdirectionalongthehyperplaneh defined for z Z d by z e Pz:= ˆv, ˆv e and Q:=Q l be the projection of z on H along ˆv so that Qz: =z Pz. Now, for x Z d, β >0, ρ>0 and L>0, we define the tilted boxes with respect to the asymptotic direction ˆv by: B β,l x:={y Z d s.t. L β <y x e <L and Qy x <ρl β } 2.2 and their front boundary by We have the following. + B β,l x:={y B β,l x s.t. y x e =L}. Proposition 8 Atypical Quenched Exit Estimate, Proposition 4. of [3]. Assume there exists α > 0 such that η α := max e U E ω0,e α <. Take M 5d +5 such that P M l is satisfied. Let β 0 /2,, β β0+ 2, and ζ 0,β 0. Then, for each γ >0 we have that where lim supl gβ0,β,ζ logpp 0,ω X TBβ,L 0 + B β,l 0 e γlβ <0, L gβ 0,β,ζ:=min{β +ζ,3β 2+d β β 0 }.

12 2 É. Bouchet, A.F. Ramírez and C. Sabot 2.3. Some results on flows The main tool that enables us to improve the results of [3] is the use of flows and maxflow min-cut theorems. We need some definitions and properties that we will detail in this section. In the following, we consider a finite directed graph G=V,E, where V is the set of vertices and E is the set of edges. For all e E, we denote by e and e the vertices that are the tail and head of the edge e the edge e goes from e to e. Definition 3. We consider a finite directed graph G=V,E. A flow from a set A V to a set Z V is a non-negative function θ:e R + such that: x A Z c, divθx=0. x A, divθx 0. x Z, divθx 0. Where the divergence operator is div:r E R V such that for all x V, divθx= θe θe. e E,e=x e E,e=x A unit flow from A to Z is a flow such that x Adivθx =. Then we have also x Z divθx=. We will need the following generalized version of the max-flow min-cut theorem. Proposition 9 Proposition of [9]. Let G=V,E be a finite directed graph. Let ce e E be a set of non-negative reals called capacities. Let x 0 be a vertex and p x x V be a set of non-negative reals. There exists a non-negative function θ:e R + such that divθ= x V p x δ x0 δ x 2.3 and if and only if for all subset K V containing x 0 we have c + K e E, θe ce 2.4 x K c p x, 2.5 where + K ={e E,e K,e K c } and c + K= e +Kce. The same is true if we restrict the condition 2.5 to the subsets K such that any y K can be reached from x 0 following a directed path in K. Wewillgivehereanideaoftheproof,thatexplainswhywecallthis resultageneralized version of the classical max-flow min-cut theorem. The complete proof can be found in [9].

13 Sharp ellipticity conditions for ballistic behavior 3 Idea of the proof. If θ satisfies 2.3 and 2.4, then divθx= e,e K,e K c θe e,e K,e K c θe= x K x K c p x. It implies 2.5 by 2.4 and positivity of θ. The reversed implication is an easy consequence of the classical max-flow min-cut theorem on finite directed graphs see, e.g., [6] Section 3.. If ce e E satisfies 2.5, we consider the new graph G=V {δ},ẽ, where Ẽ =E {x,δ,x V}. We define a new set of capacities ce e Ẽ where ce= ce for e E and cx,δ = p x. The strategy is to apply the max-flow min-cut theorem with capacities c and with source x 0 and sink δ. It gives a flow θ on G between x 0 and δ with strength x V p x and such that θ c. The function θ obtained by restriction of θ to E satisfies 2.4 and 2.3. Forthe proofof Theorem, we will considerthe orientedgraph Z d,e Z d where E Z d := {x,y Z d 2 s.t. x y =}. This graph is not finite, but we will only consider flows with compact support θe=0 for all e except in a finite subset of E Z d. We can then proceed as if the graph were finite, and use the previous definition and proposition. 3. Proof of Theorem We will prove Theorem using the atypical quenched exit estimate Proposition 8. Let us give a rough idea of the proof. We first show that the event {τ >u} is concentrated on theeventthat therandomwalkdoesnotexitaboxofside Clogu /β,foranappropriate choice of C, before time u. Now on this last event, necessarily, the walk must visit some point of this box at least N u := u/clogu d/β times. But due to Proposition 8 and the strong Markov property, the probability that this point is visited N u times is less than u u/clogud/β, for some ε>0 which depends on the choice of C. This last ε quantity tends quickly to 0, and then the dominant term bounding P 0 τ >u will be the probability to exit the box of side Clogu /β before time u. Let l S d, β >0 and M 5d+5. Assume that P M l is satisfied and that E β holds. Let us take a rotation ˆR such that ˆRe = ˆv. We fix β 5 6,, M > 0 and for simplicity we will write τ instead of τˆv. For u>, take L = Lu:= 4M d [ C L := ˆR L 2ˆv e, /β L 2ˆv e logu /β, ] d Z d.

14 4 É. Bouchet, A.F. Ramírez and C. Sabot with and Following the proof of Proposition 5. in [3], we write P 0 τ >u P 0 τ >u,t CLu τ +EF c,p 0,ω T CLu >u+pf, { F := ω Ω: t ω C Lu > { t ω A:=min u logu /β } n 0: supp x,ω T A >n x 2 As in [3], the term P 0 τ >u,t CLu τ is bounded thanks to condition P M l, and the term EF c,p 0,ω T CLu >u is bounded thanks to the strong Markov property. This part of the original proof is not modified, so we will not give more details here. It gives the existence for every γ β, of a constant c>0 such that: }. P 0 τ >u e cluγ c + logu /β +PF. 2 It only remains to show that we can find a constant C >0 such that PF Cu β for u big enough. For each ω Ω, still as in [3], there exists x 0 C Lu such that P x0,ω H x0 >T CLu 2 C Lu t ω C Lu, where for y Z d, Hy =min{n :X n =y}. It gives PF P ω Ω s.t. x 0 C Lu s.t. P x0,ω H x0 >T CLu 2logu/β C Lu. u We define for each point x C Lu a point y x, closest from x+2 Lβ ˆv e ˆv. To bound PF, we will need paths that go from x to y x with probability big enough and the atypical quenched exit estimate Proposition 8. Define: It is straightforward that N := ˆv logu 2M dˆv e. N y x x N +. The following of the proof will be developed in three parts: first, we will construct unit flows θ i,x going from {x,x+e i } to {y x,y x +e i }, for all x C Lu. Then we will construct

15 Sharp ellipticity conditions for ballistic behavior 5 paths with those flows, and use the atypical quenched exit estimate to bound PF in the case that those paths are big enough. We will conclude by bounding the probability that the paths are not big enough. 3.. Construction of the flows θ i,x We consider the oriented graph Z d,e Z d where E Z d :={x,y Z d 2 s.t. x y =}. We want to construct unit flows θ i,x going from {x,x + e i } to {y x,y x + e i }, for all x C Lu. But there are additional constraints, as we will need them to construct paths that have a probability big enough. The aim of this section is to prove the following proposition. Proposition 0. For all x C Lu, for all α,...,α 2d positive constants, there exists 2d unit flows θ i,x : E Z d R +, respectively, going from {x,x+e i } to {y x,y x +e i }, such that: e E Z d, θ i,x e αe κ i, 3. where κ i :=2 2d j= α j α i +α i+d, and αe:=α j for e of the type z,e j. Furthermore, we can construct θ i,x with a finite support, and in a way that allows to find γ and S E Z d, S independent of u, such that θ i,x eκ i γ <αe for all e S c. We will construct the θ i,x to prove their existences. For this, we need three steps. Let Bx,R be the box of Z d of center x and radius R, and B i x,r be the same box, where the vertices x and x+e i are merged and we suppress the edge between them. We note E Bx,R :={x,y E Z d Bx,R 2 } and E Bix,R := {x,y E Z d B i x,r 2 } the correspondingsetsofedges.wewillconstructaunitflowinthegraphb i x,r,e Bix,R from {x,x + e i } to B i x,r c, a unit flow in the graph B i y x,r,e Biy x,r from B i y x,r c to {y x,y x +e i }, and then connect them. At each step, we will ensure that condition 3. is fulfilled. First step. Construction of a unit flow from {x,x+e i } to B i x,r c : Lemma. Set x C Lu, and α,...,α 2d positive constants. If R maxiκi min j α j, there exists 2d unit flows θ i,x :E Bix,R R + such that: and divθ i,x = z B ix,r e E Bix,R, B i x,r δ x δ z θ i,x e αe κ i, where B i x,r={z B i x,r that has a neighbour in B i x,r c }.

16 6 É. Bouchet, A.F. Ramírez and C. Sabot The divergence condition ensures that the flow will be a unit flow, that it goes from x, and that it leaves B i x,r uniformly on the boundary of the box. Proof of Lemma. TheresultisasimpleapplicationofProposition9.We fixx C Lu and i between and 2d. Define p z = B if z B ix,r ix,r, p z =0 if z / B i x,r. To prove the result, we only have to check that K B i x,r containing x, αe e +K κ i z/ K p z, where + K ={e E Bix,R s.t. e K and e / K}. We have two cases to examine: If K B i x,r=, z/ K p z =. We then need e αe κ +K i. For K ={x}, e αe = κ +K i as we merged x and x+e i. For bigger K, we consider for all j i the paths x+ne j n N and for all j i+d the paths x+e i +ne j n N. They intersect the boundary of K in 2d+ different points, and the exit directions give us the corresponding α j, that sum to κ i. It gives that e αe κ +K i. If K B i x,r, z/ K p z <. As K contains a path from x to B i x,r, αe e +K κ i Rminjαj+α j+d κ i. It is bigger than thanks to the hypothesis on R. It gives the result. and Second step. By the same way, we construct a flow θ i,x :E Biy x,r R + such that divθ i,x = z B iy x,r e E Biy x,r, B i y x,r δ z δ yx θ i,x e αe κ i. Third step. We will join the flows on E Bix,R and E Biy x,r with simple paths, to get a flow on E Z d. Take R maxiκi min j α j, and make sure that Bx,R < αe κ i for all e E Z d always possible by taking R big enough, R depends only on the α i and the dimension. We can find Bx,R simple paths π j E Z d satisfying: j, π j connects a point of Bx,R to a point of By x,r. j, π j stays outside of Bx,R and By x,r, except from the departure and arrival points. If two paths intersect, they perform jumps in different direction after the intersection no edge is used by two paths. If x,e i is in a path, then x+e i, e i is not in any path. The number of steps of each path is close to N: there exists constants K and K 2 independent of u such that the length of π j is smaller than K N +K 2. E.g., we can use the paths π i,j page 45 of [3], and make them exit the ball Bx,R instead of {x,x+e i }. For all i, B i x,r = Bx,R and B i y x,r = By x,r as soon as R >. By construction, divθ i,x z = divθ i,x z 2 = Bx,R for any z B i x,r and z 2

17 Sharp ellipticity conditions for ballistic behavior 7 B i y x,r. We can then join the flows of the first two steps by defining a flow θ i,x e= By for all e π x,r j and 0 on all the other edges of E Z d. We have thus constructed a unit flow θ i,x on E Z d, from {x,x+e i } to {y x,y x +e i }, satisfying is satisfied on E Bix,R and E Biy x,r as R maxiκi min j α j thanks to Lemma, and outside those balls as Bx,R < αe κ i for all e E Z d. It concludes the proof of the first part of Proposition 0. As θ i,x e=0 out of the finite set E Bix,R E Biy x,r {e π j, j Bx,R }, the flow has a finite support. And as we made sure that Bx,R < αe κ i, we can take S =Bx,R By x,r and γ = to conclude the proof. κ i Bx,R 3.2. Bounds for PF We apply Proposition 0 for the α,...,α 2d of the definition of E β see.2 and.3. It gives flows θ i,x on E Z d, constructed as in the previous section. We can decompose a given θ i,x for i and x fixed in a finite set of weighted paths, each path starting from x or x+e i and arriving to y x or y x +e i. It suffices to choose a path σ where the flow is always positive, to give it a weight p σ :=min e σ θ i,x e>0 and to iterate with the new flow θe:=θ i,x e p σ e σ. The weight p σ of a path σ then satisfies: for all e E Z d, θ i,x e= σ containing e p σ. As θ i,x is a unit flow, we get σ path of θ i,x p σ =. We will use those weights in the next section, to prove that those paths are big enough with high probability. We now introduce: { F 2,i = ω Ω s.t. x C Lu, σ path of θ i,x,ω σ := ω e >u }, /M e σ where we recall that M is the parameter for condition P M, and Define F 2 = 2d i= F 2,i. } F 3 :=F 2 {ω Ω s.t. x 0 C Lu s.t. P x0,ω H x0 >T CLu 2logu/β C Lu. u We get immediately: PF PF 3 +PF c 2. It gives two new terms to bound. We start by bounding PF 3. For this we will use the same method as in [3]: on the event F 3, for all i 2d we can use a path σ of θ i,x to

18 8 É. Bouchet, A.F. Ramírez and C. Sabot join x or x+e i to y x or y x +e i. It gives: ωx 0,e i u /M min P z,ωt CLu <H x0 P x0,ωt CLu < H x0 z {y x0,y x0 +e i} 2logu/β C Lu, u where the factor ωx 0,e i corresponds to the probability of jumping from x to x+e i, in the case where the path σ starts from x+e i. As 2d i= ωx 0,e i =, it gives u /M min P z,ωt CLu <H x0 4dlogu/β C Lu, z Vy x0 u where Vy x0 :={y x0,y x0 +e i i=,...,2d }. In particular, on F 3, we can see that for u large enough Vy x0 C Lu. As a result, on F 3, we have for u large enough where min P z,ωx Tz+U z V y x0 β,l e >z e min P z,ωt CLu <H x0 z Vy x0 dlu β =e 2, u/2m U β,l :={x Z d : L β <x e <L}. From this and using the translation invariance of the measure P, we conclude that: P x 0 C Lu s.t. P x0,ω H x0 >T CLu 4dlogu/β C Lu,F 2 u P x 0 C Lu s.t. min P z,ωx Tz+Uβ e >z e e 2 z Vy x0,l 2d+ C Lu PP 0,ω X TUβ,Lu e >0 e 2 dlu β 2d+ C Lu PP 0,ω X TBβ,Lu e >0 e 2 dlu β, dlu β where the tilted box B β,lu is defined as in 2.2. We conclude with the atypical quenched exit estimate Proposition 8: there exists a constant c>0 such that for each β 0 2, one has: PF 3 c e clugβ 0,β,ζ, where gβ 0,β,ζ is defined as in Proposition 8.

19 Sharp ellipticity conditions for ballistic behavior 9 Note that for each β 5 6, there exists a β 0 2,β such that for every ζ 0, 2 one has gβ 0,β,ζ>β. Therefore, replacing L by its value, we proved that there exists c>0 such that: 3.3. Bound for PF c 2 PF 3 cu β. To conclude the bound for PF and the proof of Theorem, it only remains to control PF c 2. It is in this section that we will use the conditions that were imposed on θ i,x during the construction of the flows, as well as condition E β PF c 2 2d PF2,i c i= 2d i= x C Lu P σ path of θ i,x,ω σ u /M. As θ i,x is a unit flow, if σ path of θ i,x, ω σ u /M then: p σ ω σ u /M p σ =u /M. σ path of θ i,x σ path of θ i,x Jensen s inequality then gives: ωe θi,xe = e E Z d σ path of θ i,x ω pσ σ u /M. It allows to write: PF c 2 2d P i= x C Lu e E Z d ω θi,xe e u /M 2d i= x C Lu E e E Z d ω κiθi,xe e u κi/m. We will use the integrability given by the flows to bound the expectations. The independence of the environment gives for i and x fixed: E = e ω κiθi,xe e z Z d E e s.t. e=z = z SE e s.t. e=z ω κiθi,xe e ω κiθi,xe e z/ S E e s.t. e=z ωe κiθi,xe,

20 20 É. Bouchet, A.F. Ramírez and C. Sabot where we recall that S =Bx,R By x,r. As θ i,x satisfies 3., the ellipticity condition E β gives that each of the expectations E e s.t. e=z ω κiθi,xe e are finite. By construction S is finite and does not depend on u: z S E e s.t. e=z ω κiθi,xe e is a finite constant independent on u. It remains to deal with the case of z / S. As we chose R to get θ i,x eκ i <γ for the edges outside S, and thanks to the bounds on the number of edges with positive flow there is a finite number of paths, and each path has a bounded length, we have: z/ S E e s.t. e=z cn+c 2 ωe κiθi,xe E ωe γ, e s.t. e=0 where c and c 2 are positive constants, independent of u. Then, putting all of those bounds together, PF c 2 2d i= C C C3N 2 u κi/m x C Lu C 6 logu C7 u C8/M miniκi, 2d i= x C Lu C 4 u C5+κi/M κi where all the constants C i are positive and do not depend on u. As Remark 5 tells us that we can choose M as large as we want, we can get C8 M as small as we want. Then we can find a constant C > 0 such that PF2 c Cu β for u big enough. It concludes the proof. 4. New examples of random walks satisfying the polynomial condition 4.. Proof of Theorem 4 Consider the box B e for L=70L 3. We want to find some integer L>c 0 such that P 0 X TBe e <L L M, for some M 5d+5. We first decompose this probability according to whether the exit point of the random walk from the box B e is on the bottom or on one of the sides of the box, so that, P 0 X TBe e <L =P 0 X TBe e = L+ d P 0 X TBe e i = L+P 0 X TBe e i = L. i=2

21 Sharp ellipticity conditions for ballistic behavior 2 We will first bound the probability to exit through the sides. We do the computations for P 0 X TBe e 2 = L but the other term can be dealt with in the same way. Suppose that X TBe e 2 = L, and define n 0,...,n L the finite hitting times of new levels in direction e 2 as follows: n k :=min{n 0 s.t. X n e 2 k}. To simplify notation define ϕx:=ωx,e +d. We now choose a constant >δ>0, and we will call good point any x Z 2 such that ϕx > δ. We define p :=Pϕx > δ. Note that p does not depend on x since the environment is i.i.d., so that it depends only on δ and the law of ϕ. We now introduce the event that a great number of the X nk are good points: { C := X TBe e 2 = L and at least p } of the X nk, k 2 L L, are good. We get immediately P 0 X TBe e 2 = L=P 0 C +P 0 {X TBe e 2 = L} C c. By construction of the X nk and independence of the environment, and with Z an independent random variable following a binomial law of parameters p and L, we can bound the second term of the sum: P 0 {X TBe e 2 = L} C c P Z p 2 L exp 2 p L p L/2 2 L =exp p2 L, 2 where the last inequality is Hoeffding s inequality. It only remains to bound P 0 C. For that, we introduce the following new event C 2 :=C {X nk+ X nk =e for at least δp4 } L of the good X nk, that states that the walk goes often in direction e just after reaching a X nk that is a good point. We can then write P 0 C =P 0 C 2 +P 0 C C c 2. To bound P 0 C C c 2, we use the uniform bound ϕx > δ for good points that gives us that ωx,e >rδ on those points. Getting Z an independent random variable

22 22 É. Bouchet, A.F. Ramírez and C. Sabot following a binomial law of parameters rδ and p 2 L, it gives: P 0 C C2 P c Z δp 4 L exp pδ 2 L r 2. 2 It only remains to bound P 0 C 2. Set n + resp., n the total number of jumps in direction e resp., e before exiting the box B e. We will need a third new event that allows us to write { C 3 := n + +r } 2 n, P 0 C 2 =P 0 C 2 C 3 +P 0 C 2 C c 3. First, notice that for L big enough, C 2 C 3 =. Indeed, C implies that we exit the box B by the side x e e 2 = L. Now, since the vertical displacement of the walk before exiting the box B is e n+ n, on the event C 3 we know that this displacement is L= 35δp 2 L3 at least equal to r r+ n+. Therefore, since on C 2 the walk makes at least δp 4 moves in the direction e, on C 2 C 3 its vertical displacement before exiting the box is at least 35δpr 2r+ L 3. Since on C 2 C 3 the walk exits the box by the x e 2 = L side, we see that for L larger than L := 2+r 35δpr the event C 2 C 3 is empty. We now want to bound P 0 C 2 C3 c P 0 C 2 C3 P c 0 n + δp 4 L and n + < +r 2 n P 0 n + +n δp 4 L and n + +n 2 3+r <n. Now note that whenever we go through a vertical edge from a point x, the law of the environment tells us that it is an edge x,e with probability r +r, and x, e with probability +r. Then, defining Z as a random variable following a binomial law of parameters +r and pδ L, 4 we have the bound: P 0 C 2 C c 3 P Z 2 exp 2pδ L 3+r pδ 4 L 3+r 2 pδ L+r 2, where we need r+pδ L 23+r to apply Hoeffding s inequality in the last inequality. We can find L 2 such that this is true for L L 2.

23 Sharp ellipticity conditions for ballistic behavior 23 Choose M 5d+5. By putting all of our previous bounds together, we finally get, for all L L 2, P 0 X TBe e 2 = L exp p2 L +exp pδ 2 L r 2 2 +exp 2pδ L r 2, pδ L+r where we recall that L=70L 3, δ>0 and p=pϕx>δ. Then, for any choice of δ, we can find L 3 maxc 0,L,L 2 such that for all L L 3, d P 0 X TBe e i = L+P 0 X TBe i=2 e i = L 2L M. We now only need to bound P 0 X TBe e = L to prove P M e. We will use again the notations n + resp., n for the total number of jumps in direction e resp., e before exiting the box B. Suppose that X e T Be e = L. Then necessarily n + <n, which gives n + < n+ +n 2. As n + conditioned to n + +n follows a binomial r law of parameters +r > 2 and n+ +n, Hoeffding s inequality gives the bound: P 0 n + < n+ +n r 2 n+ +n exp 2n + +n +r 2. 2 But X TBe e = L also gives that necessarily, n L. Then P 0 X TBe e = L P 0 n + < n+ +n 2 m=l exp 2m Therefore, we can find L 4 L 3 such that for all L L 4, P 0 X TBe from where we conclude that for all L L 4, 4.2. Proof of Theorem 5 P 0 X TBe e = L 2L M, e <L L M. and n L r +r 2 2. It is classical to represent Dirichlet distributions with independent gamma random variables:if γ,...,γ N areindependent gammarandomvariableswith parametersβ,...,β N,

24 24 É. Bouchet, A.F. Ramírez and C. Sabot then γ γ γi,..., γi N is a Dirichlet random variable with parameters β,...,β N. We get a restriction property as an easy consequence of this representation see [7], pages 79 82: if ξ,...,ξ N is a Dirichlet random variable with parameters β,...,β N, for any ξ J non-empty subset of {,...,N}, the random variable j ξi j J follows a Dirichlet i J law with parameters β j j J and is independent of i J ξ i. This property will be useful in the following. We consider the box B for L=70L 3 e, and want to find some L>c 0 such that P 0 X TBe e <L L to prove P M M e. Let l i := {x Z d s.t. x e = i} and t i := min{n 0: X n l i,x n+ / l i }. We first consider the events that, when the walk arrives on l i for the first time, it gets out of it by an edge in direction e the alternative being getting out by an edge in direction e : G,i :={X ti+ X ti =e }. Atthepoint X ti,weknowthatthe walkwillgoeithertox ti +e orto X ti e.thanks ωx to the restriction property of the Dirichlet laws, we know that ti,e ωx ti,e +ωx ti, e follows a beta law of parameters β,β +β +d and is independent from the previous trajectory ofthe walkon l i. Indeed, wealreadyknowby constructionthat it is independent fromthe environmentonthe otherpoints of l i, and the restrictionpropertygivesthe independence ωx from ti,e j ωx ti,e ωx ti, e for all j,, which correspondsto the law of the previous trajectory from X ti on l i. Then P 0 G,i = β β +β +d. Now define and note that We can now write G := L G,i, i= P 0 G c L β +d β +β +d. P 0 X TBe e <L P 0 {X TBe e <L} G +L β +d β +β +d, and we only need to bound the first term of this sum. If G is satisfied, the walk cannot get out of the box B e by the lower boundary {x Zd s.t. x e = L}. Then the

25 Sharp ellipticity conditions for ballistic behavior 25 walk has to get out by one of the 2d 2 side boundaries : d P 0 {X TBe e <L} G =P 0 j=2 {X TBe e j =± L} G. On the event d j= {X T Be e j =± L} define n 0,...,n L as the finite hitting times of new levels in any direction perpendicular to e as follows: { } n k :=min n 0 s.t. max X n e j k. 2 j d Let now p= β + i +d βi and consider the event G 3 :=G {X nk+ X nk =e for at least p2 L } of the points X nk. Suppose β +d, then p Eω0,e. Consider now a random variable Z with a binomial law of parameters p and L. Using Hoeffding s inequality, we see that PG c 3 P Z p L exp p2 2 2 L. But clearly G G 3 = for L L 0 := 35p. Therefore, we have in this case P 0 d j=2 {X TBe e j =± L} G exp p2 2 L. Putting the previous bounds together, we finally get for all L L 0 : P 0 X TBe e <L L β +d +exp p2 β +β +d 2 L. Let now L be such that for all L L exp p2 2 L 2L M. The constant c 0 cf.. is increasingin β +d. Therefore,in the regionwhere β +d β, it does not depend on β +d. Call this value c 0. On the other hand, by construction, L 0 and L do not depend on β +d. Take now L 2 :=max{c 0,L 0,L }. We can then choose β +d necessarily β so that L 2 β +d β +β +d 2L M 2.

26 26 É. Bouchet, A.F. Ramírez and C. Sabot We then conclude that for this choice of β +d there exists an L c 0 such that Acknowledgements P 0 X TBe,L 2, L e <L L M. The authors would like to thank Alexander Drewitz for pointing out the ideas of the proof of Theorem 4. This work was partially supported by Fondo Nacional de Desarrollo Científico y Tecnológico grants and 4094, Iniciativa Científica Milenio NC20062 and by the ANR project MEMEMO2. References [] Berger, N., Drewitz, A. and Ramírez, A.F Effective polynomial ballisticity conditions for random walk in random environment. Comm. Pure Appl. Math MR [2] Bouchet, É Sub-ballistic random walk in Dirichlet environment. Electron. J. Probab. 8 no. 58, 25. MR [3] Campos, D. and Ramírez, A.F Ellipticity criteria for ballistic behavior of random walks in random environment. Probab. Theory Related Fields MR [4] Enriquez, N. and Sabot, C Edge oriented reinforced random walks and RWRE. C. R. Math. Acad. Sci. Paris MR [5] Enriquez, N. and Sabot, C Random walks in a Dirichlet environment. Electron. J. Probab electronic. MR [6] Lyons, R. and Peres, Y Probabilities on Trees and Networks. Cambridge: Cambridge Univ. Press. To appear. Available at rdlyons/. [7] Pemantle, R Phase transition in reinforced random walk and RWRE on trees. Ann. Probab MR [8] Rassoul-Agha, F. and Seppäläinen, T Almost sure functional central limit theorem for ballistic random walk in random environment. Ann. Inst. Henri Poincaré Probab. Stat MR [9] Sabot, C Random Dirichlet environment viewed from the particle in dimension d 3. Ann. Probab MR [0] Sabot, C. and Tournier, L. 20. Reversed Dirichlet environment and directional transience of random walks in Dirichlet environment. Ann. Inst. Henri Poincaré Probab. Stat MR [] Simenhaus, F Asymptotic direction for random walks in random environment. Ann. Inst. Henri Poincaré Probab. Stat [2] Sznitman, A.-S Slowdown estimates and central limit theorem for random walks in random environment. J. Eur. Math. Soc. JEMS MR [3] Sznitman, A.-S On a class of transient random walks in random environment. Ann. Probab MR84976 [4] Sznitman, A.-S. and Zerner, M A law of large numbers for random walks in random environment. Ann. Probab MR74289

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