ASYMPTOTIC DIRECTIONS IN RANDOM WALKS IN RANDOM ENVIRONMENT REVISITED

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1 ASYMPTOTIC DIRECTIONS IN RANDOM WALKS IN RANDOM ENVIRONMENT REVISITED A. Drewitz,2,, A.F. Ramírez,, February 6, 2009 Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Vicuña Macenna 4860, Macul, Santiago, Chile; 2 Institut für Mathemati, Technische Universität Berlin, Ser. MA 7-5, Str. des 7. Juni 36, 0623 Berlin, Germany; drewitz@math.tu-berlin.de Abstract Recently Simenhaus in [Sim07] proved that for any elliptic random wal in random environment, transience in the neighborhood of a given direction is equivalent to the a.s. existence of a deterministic asymptotic direction and to transience in any direction in the open half space defined by this asymptotic direction. Here we prove an improved version of this result and review some open problems Mathematics Subject Classification. 60K37, 60F5. Keywords. Random wal in random environment, renewal times, asymptotic directions. Introduction For each site x Z d, consider the vector ω(x) := {ω(x, e) : e Z d, e = } such that ω(x, e) (0, ) and e = ω(x, e) =. We call the set of possible values of these vectors P and define the environment ω = {ω(x) : x Z d } Ω := P Zd. We define a random wal on the random environment ω, as a random wal { : n N} with a transition probability from a site x Z d to a nearest neighbor site x + e with e = given by ω(x, e). Let us call P x,ω the law of this random wal starting from site x in the environment ω. Let P be a probability measure on Ω such that the coordinates {ω(x)} of ω are i.i.d. We call P x,ω the quenched law of the random wal in random environment (RWRE), starting from site x. Furthermore, we define the averaged (or annealed) law of the RWRE starting from x by P x := Ω P x,ω dp. In this note we discuss some aspects of RWRE related to the a.s. existence of an asymptotic direction in dimension d 2, briefly reviewing some of the open questions which have been unsolved and proving an improved version of a recent theorem of Simenhaus on the a.s. existence of an asymptotic direction. Some very fundamental and natural questions about this model remain open. Given a vector l R d \{0}, define the event A l := { n l = }. Whenever A l occurs, we say that the random wal is transient in the direction l. Let also { } l B l := inf > 0. n n Whenever B l occurs, we say that the random wal is ballistic in the direction l. We have the following open problem. Open problem.. In dimensions d 2, transience in the direction l implies ballisticity in the direction l. Partially supported by the International Research Training Group Stochastic Models of Complex Processes. Partially supported by Iniciativa Científica Milenio P F. Partially supported by Fondo Nacional de Desarrollo Científico y Tecnológico grant

2 Some partial progress related to this question has been achieved by Sznitman and Zerner [SZ99], and later by Sznitman in [Szn00, Szn0, Szn02], which we will discuss below. Under a uniform ellipticity assumption, i.e. P(ess inf min e ω(0, e) > 0) =, the following lemma, which we call Kaliow s zero-one law, was proved by Sznitman and Zerner (cf. Lemma in [SZ99]) using regeneration times. Later Zerner and Merl [ZM0] derived the corresponding result under the assumption of ellipticity only, i.e. P(min e ω(0, e) > 0) = ; cf. Proposition 3 in [ZM0]. Lemma.2 (Sznitman-Zerner). For l R d \{0}, P 0 (A l A l ) = 0 or. On the other hand, in dimension d = a zero-one law holds, i.e. P 0 (A l ) {0, }. Zerner and Merl, proved the following (see Theorem in [ZM0]). Theorem.3 (Zerner-Merl). In dimension d = 2, for l R 2 \{0}, P 0 (A l ) = 0 or. Nevertheless, we still have the following open problem. Open problem.4. In dimensions d 3, for l R d \{0}, P 0 (A l ) = 0 or. Combining Kaliow s zero-one law with the law of large numbers result of Sznitman and Zerner in [SZ99], Zerner [Zer02] proved the following theorem. Theorem.5 (Sznitman-Zerner). In dimensions d 2, there exists a direction ν S d, ν 0, and v, v 2 [0, ] such that P 0 -a.s. n n = v ν Aν v 2 ν A ν. Indeed, Theorem of [TZ04], the proof of which can be performed in the same manner with the assumption of ellipticity only instead of uniform ellipticity, implies that for e Z d with e = and there exist v e, v e [0, ] such that P 0 -a.s. P 0 (A e A e ) = (.) e n n = v e Ae v e A e. (.2) Combining this with Theorem of [Zer02] we may omit assumption (.) and still obtain (.2). Having (.2) for the elements e,..., e d of the standard basis of R d, we obtain that n /n exists P 0 -a.s. and may tae values in a set of cardinality 2 d. Employing the same argument as Goergen in p. 2 of [Goe06] we now obtain that P 0 -a.s. n /n taes two values at most. This yields Theorem.5. Whenever n / exists P 0 -a.s. we call this it the asymptotic direction and we say that a.s. an asymptotic direction exists. The existence of an asymptotic direction can already be established assuming some of the conditions introduced by Sznitman which imply ballisticity. Let γ (0, ) and l S d. The condition (T ) γ holds relative to l if for all l S d in a neighborhood of l, sup L γ log P 0 ({X TUl l < 0}) < 0, (.3) L,b,L for all b > 0, where U l,b,l = {x Z d : bl < x l < L} is a slab and T Ul,b,L = inf{n 0 : / U l,b,l} is the first exit time of this slab. On the other hand, one says that condition (T ) holds relative to l if condition (T ) γ holds relative to l for every γ (0, ). It is nown that for each γ (0, ) condition (T ) γ relative to l implies transience in the direction l and that a.s. an asymptotic direction exists which is deterministic. Also, for each γ (/2, ), condition (T ) γ relative to l implies condition (T ), which in turn implies ballisticity (see [Szn02]). One of the open problems related to condition (T ) γ is the following. Open problem.6. If (.3) is satisfied for l S d, then (T ) γ holds relative to l. 2

3 Recently in [Sim07], Simenhaus established the following theorem which gives equivalent conditions for the existence of an a.s. asymptotic direction and showing that transience in a neighborhood of a given direction implies that a.s. an asymptotic direction exists. Theorem.7 (Simenhaus). The following are equivalent: (a) There exists a non-empty open set O R d such that (b) There exists ν S d such that P 0 -a.s. P 0 (A l ) = l O. (.4) n = ν. (c) There exists ν S d such that P 0 (A l ) = for all l R d with l ν > 0. It is natural to wonder if there exists a statement analogous to Theorem.5, but related only to the existence of a possibly non-deterministic asymptotic direction. Here we answer affirmatively this question proving the following generalization of Theorem.7. Theorem.8. The following are equivalent: (a) There exists a non-empty open set O R d such that P 0 (A l A l ) = l O. (b) There exist d linearly independent vectors l,..., l d R d such that P 0 (A l A l ) = {,..., d}. (.5) (c) There exists ν S d with P 0 (A ν A ν ) = such that P 0 -a.s. n = A ν ν A ν ν. (.6) (d) There exists ν S d such that P 0 (A l A l ) = if and only if l R d is such that l ν 0. In this case, P 0 (A l A ν ) = 0 and P 0 (A l A ν ) = 0 for all l such that l ν > 0. It should be noted that Theorems.7 and.8 are interesting only in the case in which the statement of the Open Problem. is not proven to be true. Furthermore, if condition (.5) is fulfilled but (.4) is not, then if asymptotic directions exist we have to expect at least (and as it turns out at most, see also Proposition in [Sim07]) two of them. However, it is not nown whether condition (.5) can be fulfilled while (.4) is not. In fact, if the statement of the Open Problem.4 holds, then the two conditions are equivalent. Note that due to Kaliow s zero-one law, condition (d) of Theorem.8 yields a complete characterisation of P 0 (A l A l ) for all l R d. As a consequence of this result, we obtain an a priori sharper version of (c) in Theorem.7: (c ) There exists ν S d such that P 0 (A l ) = for all l R d with l ν > 0 and P 0 (A l ) = 0 if l ν 0. This observation and Theorem.3 imply that in dimension d = 2 there are at most three possibilites for the values of the set of probabilities {P 0 (A l ) : l S d }: () for all l, P 0 (A l ) = 0; (2) there exists a ν S d such that P 0 (A ν ) = while P 0 (A l ) = 0 for l ν; (3) there exists a ν S d such that P 0 (A l ) = for l such that l ν > 0 while P 0 (A l ) = 0 for l such that l ν 0. The following corollary, which can be deduced from Theorem.8, shows that nowing that there is an l such that P 0 (A l ) = and P 0 (A l ) > 0 for all l in a neighborhood of l, determines the value of P 0 (A l ) for all directions l. Corollary.9. The following are equivalent: 3

4 (a) There exists l R d and some neighborhood U(l ) such that P 0 (A l ) = and P 0 (A l ) > 0 for all l U(l ). (b) There exists ν R d such that P 0 (A l ) = for l such that l ν > 0, while P 0 (A l ) = 0 for l such that l ν 0. In particular, this shows that in Theorem.7, condition (a) can be replaced by the a priori weaer condition (a) of this corollary. In the rest of this paper we prove Theorem.8 and Corollary.9. In Section 2 we prove some preinary results needed for the proofs and in Section 3 we apply them to prove the theorem and the corollary. 2 Preinary results The implications (d) (a) (b) of Theorem.8 are obvious, so here we introduce the renewal structure and prove some preinary results needed to show that (b) (c) (d). For l R d set and for B R d define the first-exit time D l := inf{n N : l < X 0 l} D B := inf{n N : / B}; as usual, we set inf :=. We also define for l R d and s [0, ), T l s := inf{n N : l > s}. Due to their linear independence, the vectors l,... l d of Theorem.8 (b) give rise to the following 2 d cones: C σ := d ={x R d : σ (l x) 0}, σ {, } d. Furthermore, for λ (0, ] and l R d \{0} we will employ the notation C σ (λ, l) := d ={x R d : (λσ l + ( λ)l) x 0}, (2.) where the vectors defining the cone are now interpolations of the σ l with l. Note that C σ (λ, l) is a nondegenerate cone with base of finite area if and only if the vectors λσ l +( λ)l, =,..., d, are linearly independent. In particular, C σ (, l) = C σ for all σ {, } d and l. We will often choose σ such that P 0 ( d = A σ l ) > 0, which under (.5) is possible since we then have = P 0 ( d =A l A l ) = P 0 ( σ d = A σ l ) = σ P 0 ( d =A σ l ). (2.2) For a given σ {, } d which will usually be clear from the context, we will frequently consider vectors l R d satisfying the condition inf l x > 0. (2.3) x C σ S d Note here that for σ such that P 0 ( d = A σ l ) > 0 and l satisfying (2.3), the inequality P 0 (A l ) P 0 ( d = A σ l ) implies that the measure P 0 ( A l ) is well-defined. For such l we will then show the existence of a P 0 ( A l )-a.s. asymptotic direction. The strategy of our proof is based to a significant part on that of Theorem.7. We start with the following lemma which ensures that if with positive probability the random wal finally ends up in a cone, then the probability that it does so and never exits a half-space containing this cone is positive as well. Lemma 2.. Let σ {, } d and l R d such that (2.3) holds. Then P 0 ( d =A σ l ) > 0 = P 0 ( d =A σ l {D l = }) > 0. 4

5 Proof. Assume P 0 ( d = A σ l {D l = }) = 0. Then P-a.s. For y R d with l y 0 this implies P 0,ω ( d =A σ l {D l = }) = 0. (2.4) P y ( d =A σ l {D {x:l x 0} = }) = 0. (2.5) Indeed, if there existed such y with P({ω Ω P y,ω ( d = A σ l {D {x:l x 0} = }) > 0}) > 0 then for ω such that P y,ω ( d = A σ l {D {x:l x 0} = }) > 0, a random waler starting in 0 would, with positive probability with respect to P 0,ω, hit y before hitting {x : l x < 0} (due to ellipticity) and from there on finally end up in C σ without hitting {x : l x < 0}; this is a contradiction to (2.4), hence (2.5) holds. Choosing a sequence (y n ) C σ such that l y n as n we therefore get 0 = P yn ( d =A σ l {D {x:l x 0} = }) P 0 ( d =A σ l {D {x:l x l yn} = }) P 0 ( d =A σ l ) as n. To obtain the inequality we employed the translation invariance of P as well as the monotonicity of events. The following lemma will be employed to set up a renewal structure; it can in some way be seen as an analog to Lemma of [Sim07]. Lemma 2.2. Let σ {, } d such that P 0 ( d = A σ l ) > 0. Then for each l such that (2.3) holds, one has P 0 ({D Cσ(λ,l) = }) > 0 (2.6) for λ > 0 small enough. Proof. Lemma 2. implies P 0 ( d = A σ l {D l = }) > 0. Due to the ellipticity of the wal and the independence of the environment we therefore obtain P 0 ({X l > 0} ) d =A σ l (X + X ) {D l (X + X ) = } > 0, (2.7) where we name explicitly the path X + X to which the corresponding events A σ l and D l refer. Each path of the event in (2.7) is fully contained in C σ (λ, l) for λ > 0 small enough. Thus, the continuity from above of P 0 yields P 0 ({D Cσ(λ,l) = } {X l > 0} ) d =A σ l (X + X ) {D l (X + X ) = } > 0 (2.8) for all λ > 0 small enough. Employing Lemma 2.2, for σ {, } d with P 0 ( d = A σ l ) > 0 as in [Sim07] we can introduce a cone renewal structure, where we choose l R d such that (2.3) is fulfilled and the cone to wor with is C l := C σ (λ, l), where we fixed λ > 0 small enough as in the statement of Lemma 2.2. Note that for fixed l the set C σ (λ, l) is indeed a cone as long as λ > 0 is chosen small enough (since the defining vectors in (2.) are linearly independent). We define and inductively for : S l 0 := T l 0, R l 0 := D XS l 0 +C l θ S l 0 + S l 0, M l 0 := max{ l : 0 n R l 0} S l := T l M l, R l := D XS l +C l θ S l + S l, M l := max{ l : 0 n R l }, where for x Z d by x + C l we denote the cone C l shifted such that its apex lies at x. Furthermore, set K l := inf{ N : S l <, R l = } as well as τ l := S l K l, 5

6 i.e. τ l is the first time at which the wal reaches a new maximum in direction l and never exits the cone C l shifted to X τ l. We define inductively the sequence of cone renewal times with respect to C l by τ l := τ l (X +τ l ) + τ l for 2. The following lemma shows that under the conditions of Lemma 2.2 the sequence τ l is well-defined on A l. It can be seen as an analog to Proposition 2 of [Sim07]. Lemma 2.3. Let σ {, } d such that P 0 ( d = A σ l ) > 0 and choose l and λ such that (2.3) and (2.6) hold. Then P 0 ( A l )-a.s. one has K l <. Proof. Employing Lemma 2.2, the proof taes advantage of standard renewal arguments and is analogous to the proof of Proposition 2 in [Sim07] or Proposition.2 in [SZ99]. Lemma 2.4. Let σ {, } d such that P 0 ( d = A σ l ) > 0 and choose l and λ such that (2.3) and (2.6) hold. Then ((X τ l, τ l ),..., (X (τ l + ) τ l +, τ l + τ l )),... are independent under P 0( A l ) and for, ((X (τ l + ) τ + l Xl τ ), τ+ l τ l ) under P 0( A l ) is distributed lie (X τ l, τ) l under P 0 ( {D Cl = }). Proof. The proof is analogous to the proof of Corollary.5 in [SZ99]. The following lemma has been derived in Simenhaus thesis [Sim08] (Lemma 2 in there). Here we state it and prove it under a slightly weaer assumption. Lemma 2.5. Let σ {, } d such that P 0 ( d = A σ l ) > 0 and choose l Z d and λ such that (2.3) and (2.6) hold and the g.c.d. of the coordinates of l is. Then and l {D Cl = }) = P 0 ({D Cl = } A l ) i P 0 ({Ti l <, X T l = i}) < i l is well-defined. {D Cl = }) (2.9) Remar 2.6. A fundamental consequence of woring with the cone renewal structure instead of woring with slabs is the existence of (2.9), see Proposition (2.7) also. Proof. The proof leans on the proof of Lemma in [TZ04] which is due to Zerner. Due to the strong Marov property and the independence and translation invariance of the environment we have for i > 0 : P 0 ({ : X τ l l = i} A l ) = EP 0,ω ({Ti l <, X T l i = x, D Cl +X T θ l T l i i = }) x Z d,l x=i = x Z d,l x=i EP 0,ω ({Ti l <, X T l i = x})p x,ω ({D Cl +x = }) = P 0 ({T l i <, X T l i l = i})p 0 ({D Cl = }). (2.0) At the same time using {τ l < } = A l, a fact which is proven similarly to Proposition.2 of [SZ99], we compute P 0({ : X τ l i l = i} A l ) = i P 0 ({ 2 : X τ l l = i} A l ) = i n i n i = = n P 0 ({ 2 : X τ l l = i} {X τ l l = n} A l ) P 0 ({ 2 : (X τ l ) l = i n} {X τ l l = n} A l ) P 0 ({ 2 : (X τ l ) l = i n} A l )P 0 ({X τ l l = n} A l ), (2.) 6

7 where to obtain the last equality we too advantage of Lemma 2.4. Blacwell s renewal theorem in combination with Lemma 2.4 now yields and thus (2.) implies P 0({ 2 : (X τ l i ) l = i n} A l ) = l {D Cl = }) P 0({ : X τ l i l = i} A l ) = l {D Cl = }). Therefore, taing into consideration (2.0) we infer l {D Cl = }) = P 0 ({D Cl = } A l ) i P 0 ({Ti l <, X (2.2) T l = i}). i l It remains to show that the right hand side of (2.2) is finite. Writing l max := max{ l,..., l d } for the maximum of the absolute values of the coordinates of l we have +l max i= P 0 ({T l i <, X T l i l = i}) +l max i= P 0 ({T l i <, X T l l = i}) P 0 (A l ), N, where the first inequality follows since {X T l l = i} {X T l i l = i} for all N and i {,..., + l max }. This now yields i P 0 ({Ti l <, X T l = i}) i l l maxp 0 (A l ) > 0, whence due to (2.2) we obtain l {D Cl = }) <. (2.3) Since on {D Cl = } there exists a constant C > 0 such that X τ l CX τ l l, we infer as a direct consequence of (2.3) that (2.9) is well-defined. We can now employ the above renewal structure to obtain an a.s. constant asymptotic direction on A l. Proposition 2.7. Let σ {, } d such that P 0 ( d = A σ l ) > 0 and choose l Z d and λ such that (2.3) and (2.6) hold and the g.c.d of the coordinates of l is. Then P 0 ( A l )-a.s. n = E 0(X {D τ l C l = }) {D Cl = }). Remar 2.8. In particular, this proposition implies that the it does not depend on the particular choice of l nor λ (for λ sufficiently small). Note that the independence of l stems from the fact that if l, l 2 satisfy (2.3) we have P 0 (A l A l2 ) > 0. Proof. Due to Lemmas 2.2 to 2.5 we may apply the law of large numbers to the sequence (X τ l ) N yielding and hence X τ l E 0(X τ l {D Cl = }) P 0 ( A l ) a.s.,, X τ l X τ l E 0(X {D τ l C l = }) {D Cl = }) P 0 ( A l ) a.s.,. Using standard methods to estimate the intermediate terms (cf. p. 9 in [Sim07]) one obtains n = E 0(X {D τ l C l = }) {D Cl = }) P 0 ( A l ) a.s. The following two results will be needed to obtain results about transience in directions orthogonal to the asymptotic direction. 7

8 Lemma 2.9. Let (Y n ) n N be an i.i.d. sequence on some probability space (X, F, P ) with expectation EY = 0 and variance EY 2 (0, ]. Then, for S n := n = Y we have P ({ inf n S n = }) = P ({ sup n S n = }) =. Proof. We only prove P ({ inf n S n = }) =, the remaining equality is proved in an anolog way. Setting ε := ( ess inf Y /2) one can show for all x R, using the strong Marov property at the entrance times of S n to the interval [x, x + ε], that P ({ inf n S n [x, x + ε]}) = 0. This then implies P ({ inf n S n = ± }) =. But Kesten s result in [Kes75] yields inf n S n /n > 0 P ( { inf n S n = })-a.s., while by the strong law of large numbers we have n S n /n = 0 P -a.s. This yields P ({ inf n S n = }) = 0 and hence finishes the proof. Lemma 2.0. Let l R d such that P 0 ({ n / = l}) > 0. (2.4) Then, for l R d such that l l = 0 one has P 0 ((A l A l ) A l ) = 0. Proof. We choose a basis l,..., l d of R d and σ such that l is contained in the interior of the cone C σ corresponding to l,..., l d and (2.3) is satisfied. Furthermore, by (2.4) and Lemma 2.2 we may choose λ such that condition (2.6) is satisfied for the corresponding cone C σ (λ, l). Lemma 2.3 yields that the sequence (τ l ) N is well defined and Lemmas 2.4 and 2.5 yield that under P 0 ( A l ) the sequence ((X τ l 2 ) l, (X τ l 3 2 ) l,... ) is i.i.d. with expectation 0, the latter being due to the validity of Lemma 2.5 as well as (.6) and l l = 0. Indeed, Proposition 2.7 yields l {D Cσ(λ,l) = }) = {D Cσ(λ,l) = }) X τ l X τ l } {{ } =l l = 0 P 0 ( A l ) a.s. Applying Lemma 2.9 to the sequence ((X τ l 2 X τ l ) l, (X τ l 3 X τ l 2 ) l,... ) yields P 0 ((A l A l ) A l ) = 0. 3 Proof of Theorem.8 and Corollary.9 3. Proof of Theorem.8 We first prove that condition (b) implies (c). Note that due to Lemma 2.0 and (2.3), we obtain P 0 ({ n / σ C σ }) = 0. We now choose σ such that P 0 ( d = A σ l ) > 0 and observe l {x R d : l x > 0} = int σ σ C σ ; here, with int we denote the interior of a set and the union is taen over all vectors l Z d that satisfy (2.3) and for which the g.c.d. of the coordinates of l is. Hence, letting l vary over all such vectors, Proposition 2.7 yields P 0 ( σ σ d = A σ l )-a.s. n = E 0(X {D τ l C l = }) =: ν, (3.) {D Cl = }) which due to Remar 2.8 is independent of the respective l chosen. Now if P 0 ( σ σ d = A σ l ) = this finishes the proof and the result is equivalent to Theorem.7 obtained in [Sim07]. Thus, assume P 0 ( σ σ d = A σ l ) (0, ). (3.2) In the same manner as before we obtain for any l Z d with coordinates of g.c.d. and satisfying (2.3) with σ replaced by σ n = E 0(X τ {D l Cl = }) (3.3) = }) {D Cl P 0 ( d = A σ l )-a.s. with hopefully self-explaining notations. Now Proposition of [Sim07] states that if two elements ν ν of S d occur with positive probability each with respect to P 0 as asymptotic directions, then ν = ν. Thus, (3.) to (3.3) imply that the it in (3.3) equals ν, and (3.3) holds P 0 ( A ν )-a.s. This yields (c). Now with respect to the implication (c) (d) note that the only thing that is not obvious at a first glance is that l ν = 0 implies P 0 (A l A l ) = 0. However, Lemma 2.0 yields P 0 ((A l A l ) (A ν A ν )) = 0 which due to P 0 (A ν A ν ) = yields the desired result. 8

9 3.2 Proof of Corollary.9 We only have to prove (a) (b). Given (a), Theorem.8 yields the existence of ν S d such that P 0 (A ν A ν ) = (3.4) and (.6) holds. Now if l ν 0 then P 0 (A ν A l ) = or P 0 (A ν A l ) =, respectively, and hence P 0 (A ν ) = or P 0 (A ν ) =, which due to Theorem.7 finishes the proof. Thus, assume l ν = 0 (3.5) from now on. Then Lemma 2.0 yields P 0 ((A l A l ) (A ν A ν )) = 0 which due to (3.4) implies P 0 (A l A l ) = 0, a contradiction to assumption (a). Acnowledgments. We than François Simenhaus for reading a preinary version of this paper and for maing several useful comments on it. 9

10 References [Goe06] Laurent Goergen. Limit velocity and zero-one laws for diffusions in random environment. Ann. Appl. Probab., 6(3):086 23, [Kes75] Harry Kesten. Sums of stationary sequences cannot grow slower than linearly. Proc. Am. Math. Soc., 49:205 2, 975. [Sim07] François Simenhaus. Asymptotic direction for random wals in random environments. Ann. Inst. H. Poincaré, 43(6):75 76, [Sim08] François Simenhaus. Marches Aléatoires en Milieux Aléatoires Étude de quelques Modèles Multidimensionnels. PhD thesis, Université Paris 7 - Denis Diderot, [SZ99] Alain-Sol Sznitman and Martin Zerner. A law of large numbers for random wals in random environment. Ann. Probab., 27(4):85 869, 999. [Szn00] Alain-Sol Sznitman. Slowdown estimates and central it theorem for random wals in random environment. J. Eur. Math. Soc. (JEMS), 2(2):93 43, [Szn0] Alain-Sol Sznitman. On a class of transient random wals in random environment. Ann. Probab., 29(2): , 200. [Szn02] Alain-Sol Sznitman. An effective criterion for ballistic behavior of random wals in random environment. Probab. Theory Related Fields, 22(4): , [TZ04] Simon Tavaré and Ofer Zeitouni. Lectures on probability theory and statistics, volume 837 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, Lectures from the 3st Summer School on Probability Theory held in Saint-Flour, July 8 25, 200, Edited by Jean Picard. [Zer02] Martin P. W. Zerner. A non-ballistic law of large numbers for random wals in i.i.d. random environment. Electron. Comm. Probab., 7:9 97 (electronic), [ZM0] Martin P. W. Zerner and Franz Merl. A zero-one law for planar random wals in random environment. Ann. Probab., 29(4):76 732,