Collisions of several walkers in recurrent random environments
CCOLLISIONS OF SEVERAL WALKERS IN RECURRENT RANDOMENVIRONMENTS
ALEXIS DEVULDER, NINA GANTERT, AND FRANC¸ OISE P`ENE
Abstract.
We consider d independent walkers on Z , m of them performing simplesymmetric random walk and r = d − m of them performing recurrent RWRE (Sinaiwalk), in I independent random environments. We show that the product is recurrent,almost surely, if and only if m ≤ m = d = 2. In the transient case with r ≥ m = 2and r ≥ I = 1. In particular, while I does not have an influence for the recurrence ortransience, it does play a role for the probability to have infinitely many meetings. Toobtain these statements, we prove two subtle localization results for a single walker in arecurrent random environment, which are of independent interest. Introduction and statement of the main results
Recurrence and transience of products of simple symmetric random walks on Z d is well-known since the works of P´olya [P21]. If the product of several walks is transient, one mayask if they meet infinitely often. It is also well-known and goes back to Dvoretzky andErd¨os, see ([DE51], p. 367) that 3 independent simple symmetric random walks (SRW)in dimension 1 meet infinitely often almost surely while 4 walks meet only finitely often,almost surely. In fact, P´olya’s original interest in recurrence/transience of simple randomwalk came from a question about collisions of two independent walkers on the same grid,see [P84], “Two incidents”.The classical topic of meetings/collisions of two or more walkers walking on the samegraph has found recent interest, see [KP04], [BSP12], where the grid is replaced by moregeneral graphs. It is well-known that if a graph is recurrent for simple random walk,two independent walkers do not necessarily meet infinitely often, see [KP04]. Since ona transitive recurrent graph, two independent walkers do meet infinitely often, almostsurely, see [KP04], the “infinite collision property” describes how far the recurrent graphis from being transitive. For motivation from physics, see [CC12].We investigate this question for products of recurrent random walks in random environ-ment (RWRE) and of simple symmetric random walks on Z . It is known already that,for any n , a product of n independent RWRE in n i.i.d. recurrent random environmentsis recurrent, see [Z01], and that n independent walkers in the same recurrent randomenvironment meet infinitely often in the origin, see [GKP14]. Here, we consider severalwalkers each one performing either a Sinai walk or a simple symmetric random walk, withthe additional twist that not all Sinai walkers are necessarily using the same environment. Date : July 12, 2018. a r X i v : . [ m a t h . P R ] J u l ALEXIS DEVULDER, NINA GANTERT, AND FRANC¸ OISE P`ENE
Let d, m, r be nonnegative integers such that m + r = d ≥
1. We consider d walkers, m of them performing SRW S (1) , ..., S ( m ) and the r others performing random walks Z (1) , ..., Z ( r ) in I independent random environments, with I ≤ r . More precisely, weconsider r collections of i.i.d. random variables ω (1) := (cid:0) ω (1) x (cid:1) x ∈ Z , . . . , ω ( r ) := (cid:0) ω ( r ) x (cid:1) x ∈ Z ,taking values in (0 ,
1) and defined on the same probability space (Ω , F , P ), such that ω (1) , ..., ω ( I ) are independent and such that the others are exact copies of some of these I collections, i.e., for every j ∈ { I + 1 , ..., r } , there exists an index J j ∈ { , ..., I } such that ω ( j ) ≡ ω ( J j ) . A realization of ω := (cid:0) ω (1) , ..., ω ( r ) (cid:1) will be called an environment . Recallthat we denote by P the law of the environment ω. We set Y n := (cid:0) S (1) n , ..., S ( m ) n , Z (1) n , ..., Z ( r ) n (cid:1) , n ∈ N , and make the following assumptions. Given ω = (cid:0) ω (1) , ..., ω ( r ) (cid:1) and x ∈ Z d , under P xω , S (1) , ..., S ( m ) , Z (1) , ..., Z ( r ) are independent Markov chains such that P xω ( Y = x ) = 1 andfor all y ∈ Z and n ∈ N , P xω (cid:2) S ( i ) n +1 = y + 1 (cid:12)(cid:12) S ( i ) n = y (cid:3) = 12 = P xω (cid:2) S ( i ) n +1 = y − (cid:12)(cid:12) S ( i ) n = y (cid:3) , i ∈ { , ..., m } , (1) P xω (cid:2) Z ( j ) n +1 = y + 1 (cid:12)(cid:12) Z ( j ) n = y (cid:3) = ω ( j ) y = 1 − P xω (cid:2) Z ( j ) n +1 = y − (cid:12)(cid:12) Z ( j ) n = y (cid:3) , j ∈ { , ..., r } . (2)We set S ( i ) := (cid:0) S ( i ) n (cid:1) n and Z ( j ) := (cid:0) Z ( j ) n (cid:1) n for every i ∈ { , ..., m } and every j ∈ { , ..., r } .Note that, for every j , Z ( j ) = (cid:0) Z ( j ) n (cid:1) n is a random walk on Z in the environment ω ( j ) , andthat the S ( i ) ’s are independent SRW, independent of the Z ( j ) ’s and of their environments.We call P ω := P ω the quenched law . Here and in the sequel we write 0 for the origin in Z d . We also define the annealed law as follows: P [ · ] := (cid:90) P ω [ · ] P (d ω ) . Setting ρ ( j ) k := − ω ( j ) k ω ( j ) k for j ∈ { , ..., r } and k ∈ Z , we assume moreover that there exists ε ∈ (0 , /
2) such that for every j ∈ { , ..., r } , P (cid:2) ω ( j )0 ∈ [ ε , − ε ] (cid:3) = 1 , E (cid:2) log ρ ( j )0 (cid:3) = 0 , σ j := E (cid:2) (log ρ ( j )0 ) (cid:3) > , (3)where E is the expectation with respect to P . Under these assumptions, the Z ( j ) areRWRE, often called Sinai’s walks due to the famous result of [S82]. Solomon [S75] provedthe recurrence of Z ( j ) for P -almost every environment. We stress in particular that theassumption σ j > Y := ( Y n ) n . Recurrence of Y meansthat S (1) , ...., S ( m ) , Z (1) , ..., Z ( r ) meet simultaneously at 0 infinitely often. As explainedpreviously, this result is known for SRW (i.e. if m = d ) since [P21] and more recentlyfor RWRE (i.e. if r = d , that is, if m = 0) in the case where the environments ω ( j ) areindependent (i.e. I = r = d , see [Z01, GKP14]) and in the case where the environment ω ( j ) is the same for all the RWRE (i.e. r = d, I = 1, see [GKP14]). See also [Ga13] forrelated results. OLLISIONS OF SEVERAL WALKERS IN RECURRENT RANDOM ENVIRONMENTS 3
Theorem 1.1. If m ≤ , or if m = d = 2 , then, for P -almost every ω , the random walk Y is recurrent with respect to P ω . Otherwise, for P -almost every ω , the random walk Y is transient with respect to P ω . In particular, a product of two recurrent RWRE and one SRW is recurrent, while a productof two SRW and one recurrent RWRE is transient.When Y is transient, a natural question is the study of the simultaneous meetings (i.e.,collisions) of S (1) , ...., S ( m ) , Z (1) , ..., Z ( r ) . That is, we would like to extend the results of[P21, DE51] to the case in which some of the random walks are in random environments(when r ≥ r = 0, the number of collisions is, by [P21, DE51],almost surely infinite if m ≤ m ≥
4. Interestingly,compared to Theorem 1.1, the behaviour depends on whether I = 1 (when the RWREare all in the same environment) or I ≥ Theorem 1.2.
We distinguish the following different cases. (i) If m ≥ and r ≥ , then, for P -almost every environment ω , P ω (cid:2) S (1) n = S (2) n = S (3) n = Z (1) n infinitely often (cid:3) = 0 , i.e. almost surely, the walks S (1) , S (2) , S (3) , Z (1) meet simultaneously only a finitenumber of times. A fortiori, S (1) , . . . , S ( m ) , Z (1) , . . . Z ( r ) also meet simultaneouslyonly a finite number of times. (ii) If m = 2 and r ≥ I = 1 , then for P -almost every environment ω , P ω (cid:2) S (1) n = S (2) n = Z (1) n = ... = Z ( r ) n infinitely often (cid:3) = 1 , i.e. almost surely, the walks S (1) , S (2) , Z (1) , ..., Z ( r ) meet simultaneously infinitelyoften. (iii) If m = 2 and r ≥ I ≥ , then for P -almost every environment ω , P ω (cid:2) S (1) n = S (2) n = Z (1) n = Z (2) n infinitely often (cid:3) = 0 , i.e. almost surely, the walks S (1) , S (2) , Z (1) , Z (2) , and a fortiori the walks S (1) , S (2) , Z (1) , ..., Z ( r ) , meet simultaneously only a finite number of times. This last result can be summarized in the following manner. Assume that r ≥ Y is transient (i.e. m ≥ r ≥ S (1) , ..., S ( m ) , Z (1) , ..., Z ( r ) meet simultaneouslyinfinitely often if and only if m = 2 and I = 1. Hence our results cover collisions ofan arbitrary number of random walks in equal or independent random (or deterministic)recurrent environments. Remark 1.3.
The results of Theorem 1.2 remain true if the simple random walks arereplaced by random walks on Z with i.i.d. centered increments with finite and strictlypositive variance. However, we write the proof of this theorem only in the case of SRW tokeep the proof more readable and less technical. ALEXIS DEVULDER, NINA GANTERT, AND FRANC¸ OISE P`ENE
The case of transient RWRE in the same subballistic random environment is investigatedin [DGP18] (in preparation).In order to demonstrate Theorem 1.2, we prove the two following propositions. The firstone deals with two independent recurrent RWRE in two independent environments.
Proposition 1.4.
Assume r ≥ I ≥ . For every ε > , P (cid:2) Z (1) n = Z (2) n (cid:3) = O ((log n ) − ε ) . The second proposition deals with r independent recurrent RWRE in the same environ-ment. Proposition 1.5.
Assume r > I = 1 . For P -almost every ω , there exists c ( ω ) > suchthat, for every ( y , ..., y r ) ∈ [(2 Z ) r ∪ (2 Z + 1) r ] , we have lim sup N → + ∞ N N (cid:88) n =1 n (cid:88) k ∈ Z r (cid:89) j =1 P y j ω [ Z ( j ) n = k ] ≥ c ( ω ) . These two propositions are based on two new localization results for recurrent RWRE,which are of independent interest. These two localization results use the potential of theenvironment (see (5)) and its valleys , these quantities were introduced by Sinai in [S82]and are crucial for the investigation of the RWRE.In the first one, stated in Proposition 4.5 and used to prove Proposition 1.4, we localize arecurrent RWRE at time n with (annealed) probability 1 − (log n ) − ε for ε >
0, whereasprevious localization results for such RWRE were with probability 1 − o (1) (see [S82],[G84], [KTT89], [BF08] and [F15]), or with probability 1 − C (cid:0) log log log n log log n (cid:1) / for some C > N ∈ N , with high probability on ω (for P ), the quenchedprobability P ω [ Z n = b ( N )] is larger than a positive constant, uniformly for any even n ∈ [ N − ε , N ] for some ε >
0, where b ( N ) is the (even) bottom of some valley of thepotential V of a recurrent RWRE Z (defined in (77)). In order to get this uniformprobability estimate, we use a method different from that of previous localization results,based on a coupling between recurrent RWRE.The article is organized as follows. In Section 2, we give an estimate on the returnprobability of recurrent RWRE, see Proposition 2.1, which is of independent interest. Ourmain results for direct products of walks are proved in Section 3. The proofs concerningthe simultaneous meetings of random walks are based on the above-mentioned two keylocalization results for recurrent RWRE, proved in Sections 4 and 5.2. A return probability estimate for the rwre
We consider a recurrent one dimensional RWRE Z = ( Z n ) n in the random environment ω = ( ω x ) x ∈ Z , where the ω x ∈ (0 , x ∈ Z , are i.i.d. (that is, Z = 0 and (2) is satisfied OLLISIONS OF SEVERAL WALKERS IN RECURRENT RANDOM ENVIRONMENTS 5 with Z and ω instead of Z ( j ) and ω ( j ) ). We assume the existence of ε ∈ (0 , /
2) suchthat P [ ω ∈ [ ε , − ε ]] = 1 , E [log ρ ] = 0 , E [(log ρ ) ] > , (4)where ρ k := − ω k ω k , k ∈ Z . The following result completes [GKP14, Theorem 1.1] whichsays that, for every 0 ≤ ϑ <
1, we have for P -almost every environment ω , (cid:88) n ≥ P ω [ Z n = 0] n ϑ = ∞ . Proposition 2.1.
For P -almost every environment ω , (cid:88) n ≥ P ω [ Z n = 0] n < ∞ . Before proving this result, we introduce some more notations. First, let τ ( x ) := inf { n ≥ Z n = x } , x ∈ Z . In words, τ ( x ) is the hitting time of the site x by the RWRE Z . As usual, we considerthe potential V , which is a function of the environment ω and is defined on Z as follows: V ( x ) := (cid:80) xi =1 log − ω i ω i if x > , x = 0 , − (cid:80) i = x +1 log − ω i ω i if x < . (5)The potential is useful since it relates to the description of the RWRE as an electricnetwork. It can be used to estimate ruin probabilities for the RWRE. In particular, wehave (see e.g. [Z01, (2.1.4)] and [D14, Lemma 2.2] coming from [Z01, p. 250]), P bω [ τ ( c ) < τ ( a )] = (cid:18) b − (cid:88) j = a e V ( j ) (cid:19)(cid:18) c − (cid:88) j = a e V ( j ) (cid:19) − , a < b < c (6)and, recalling ε from (3) and (4), E bω [ τ ( a ) ∧ τ ( c )] ≤ ε − ( c − a ) exp (cid:104) max a ≤ (cid:96) ≤ k ≤ c − k ≥ b (cid:0) V ( k ) − V ( (cid:96) ) (cid:1)(cid:105) , a < b < c , (7)where E bω denotes the expectation with respect to P bω and u ∧ v := min( u, v ), ( u, v ) ∈ R .For symmetry reasons, we also have E bω [ τ ( a ) ∧ τ ( c )] ≤ ε − ( c − a ) exp (cid:104) max a ≤ (cid:96) ≤ k ≤ c − , (cid:96) ≤ b − (cid:0) V ( (cid:96) ) − V ( k ) (cid:1)(cid:105) , a < b < c . (8)Moreover, we have, for k ≥ P bω [ τ ( c ) < k ] ≤ k exp (cid:18) min (cid:96) ∈ [ b,c − V ( (cid:96) ) − V ( c − (cid:19) , b < c , (9)and by symmetry, we get (similarly as in Shi and Zindy [SZ07], eq. (2.5) but with someslight differences for the values of (cid:96) ) P bω [ τ ( a ) < k ] ≤ k exp (cid:18) min (cid:96) ∈ [ a,b − V ( (cid:96) ) − V ( a ) (cid:19) , a < b . (10) ALEXIS DEVULDER, NINA GANTERT, AND FRANC¸ OISE P`ENE
Lemma 2.2.
Let γ > . For P -almost every ω , there exists N ( ω ) such that for every n ≥ N ( ω ) , n − γ ≤ max k ∈{ ,...,n } V ( k ) ≤ n + γ , − n + γ ≤ min k ∈{ ,...,n } V ( k ) ≤ − n − γ , and such that the same inequalities hold with {− n, . . . , } instead of { , . . . , n } .Proof. Observe that it is enough to prove that P -almost surely, n − γ ≤ max ≤ k ≤ n V ( k ) ≤ n + γ (11)if n is large enough (up to a change of log ρ i in − log ρ i , in log ρ − i or in − log ρ − i ). Thefirst inequality of (11) is given by [H65, Theorem 2]. The second inequality of (11) isa consequence of the law of iterated logarithm for V , as explained in ([C01], end of p.248). (cid:4) Proof of Proposition 2.1.
Let η ∈ (0 ,
1) and n ≥
2. We define z + := inf { y ≥ V ( y ) ≤ − (log n ) − η } , z − := sup { y ≤ − V ( y ) ≤ − (log n ) − η } . Due to the previous lemma, choosing γ small enough, we have that P -almost surely, if n is large enough, the following inequalities hold: | z ± | ≤ (log n ) − η z − ≤ i,j ≤ z + ( V ( i ) − V ( j )) ≤ (log n ) − η/ . (12)We have by the strong Markov property, P ω [ Z n = 0] ≤ P ω [ τ ( z + ) > n, τ ( z − ) > n ] + n (cid:88) k =0 P ω [ τ ( z + ) = k ] P z + ω [ Z n − k = 0]+ n (cid:88) k =0 P ω [ τ ( z − ) = k ] P z − ω [ Z n − k = 0] . (13)Recall that, given ω , the Markov chain Z is an electrical network where, for every x ∈ Z ,the conductance of the bond ( x, x + 1) is C ( x,x +1) = e − V ( x ) (in the sense of Doyle andSnell [DS84]). In particular, the reversible measure µ ω (unique up to a multiplication bya constant) is given by µ ω ( x ) := e − V ( x ) + e − V ( x − , z ∈ Z . (14)So we have P z ± ω [ Z n − k = 0] = P ω [ Z n − k = z ± ] µ ω (0) µ ω ( z ± ) ≤ µ ω (0) µ ω ( z ± ) = e − V (0) + e − V ( − e − V ( z ± ) + e − V ( z ± − ≤ e − V (0) + e − V ( − e − V ( z ± ) ≤ (cid:0) e − V (0) + e − V ( − (cid:1) exp (cid:2) − (log n ) − η (cid:3) . Hence, n (cid:88) k =0 P ω [ τ ( z ± ) = k ] P z ± ω [ Z n − k = 0] ≤ (cid:0) e − V (0) + e − V ( − (cid:1) exp (cid:2) − (log n ) − η (cid:3) . (15)Moreover we have due to (7) and to Markov’s inequality, P ω [ τ ( z + ) > n, τ ( z − ) > n ] ≤ n − E ω [ τ ( z + ) ∧ τ ( z − )] OLLISIONS OF SEVERAL WALKERS IN RECURRENT RANDOM ENVIRONMENTS 7 ≤ n − ε − ( z + − z − ) exp (cid:104) max z − ≤ (cid:96) ≤ k ≤ z + − [ V ( k ) − V ( (cid:96) )] (cid:105) . Now using (12), P -almost surely, we have P ω [ τ ( z + ) > n, τ ( z − ) > n ] ≤ ε − n − (log n ) − η exp (cid:2) (log n ) − η/ (cid:3) for every n large enough. This combined with (13), (15) and e − V ( − ≤ ε − gives P -almostsurely for large n P ω [ Z n = 0] ≤ ε − exp (cid:2) − (log n ) − η (cid:3) . Consequently, (cid:80) n ≥ P ω [ Z n =0] n < ∞ P -almost surely, which ends the proof of Proposition2.1. (cid:4) Direct product of Walks
We start with a proof of Theorem 1.1. With a slight abuse of notation, we will write 0for the origin in Z k , whatever k is. Proof.
1. If m ≥ r = 0, then ( Y n ) n is a product of m independent simple randomwalks on Z . It is well-known that it is recurrent if m ∈ { , } , and transient if m ≥ G n ) n ,( G n ) n is recurrent if and only if (cid:88) n ≥ P x [ G n = 0] = ∞ , (16)where x is one of the states of the Markov chain.2. If m ≥ r ≥
1, then the 3-tuple of the three first coordinates of ( Y n ) n is (cid:0) S (1) n , S (2) n , S (3) n (cid:1) n which is a product of 3 independent simple random walks on Z , henceis transient. So ( Y n ) n is transient for P -almost every ω .3. If m = 2 and r ≥
1, then applying the local limit theorem (see e.g. Lawler and Limic[LL10] Prop. 2.5.3) for S (1) and S (2) for n ∈ N ∗ , P ω [ Y n = 0] = (cid:89) i =1 P (cid:2) S ( i ) n = 0 (cid:3) r (cid:89) j =1 P ω ( j ) (cid:2) Z ( j ) n = 0 (cid:3) ≤ cn P ω (1) (cid:2) Z (1) n = 0 (cid:3) , where c > (cid:80) ∞ n =0 P ω [ Y n = 0] < ∞ for P -almost ω . Hence, (using the Borel-Cantelli Lemma or (16)), ( Y n ) n is P -almost surelytransient.4. We now assume m ∈ { , } . We choose some δ ∈ (0 , /
5) such that 3 δr < − δ . Wedenote by (cid:98) x (cid:99) the integer part of x for x ∈ R . For L ∈ N , we have (cid:88) n ≥ P ω [ Y n = 0] ≥ (cid:98) e (1 − δ ) L (cid:99) (cid:88) n = (cid:4) e (1 − δ ) L (cid:5) +1 P ω [ Y n = 0]= (cid:98) e (1 − δ ) L (cid:99) (cid:88) n = (cid:4) e (1 − δ ) L (cid:5) +1 P (cid:2) S n = 0 (cid:3) m r (cid:89) j =1 P ω ( j ) (cid:2) Z ( j )2 n = 0 (cid:3) . ALEXIS DEVULDER, NINA GANTERT, AND FRANC¸ OISE P`ENE
Due to [GKP14] (Propositions 3.2, 3.4 and (3.22)), since δ ∈ (0 , / C ( δ ) > L, δ )) L ∈ N of elements of F (that is, depending only on ω ) such that P (cid:34) (cid:92) N ≥ (cid:91) L ≥ N Γ( L, δ ) (cid:35) = 1 (17)and such that, for every L ∈ N , on Γ( L, δ ), we have ∀ i ∈ { , . . . , r } , ∀ k i ∈ (cid:8) (cid:98) e δL (cid:99) + 1 , · · · , (cid:98) e (1 − δ ) L (cid:99) (cid:9) , P ω ( i ) (cid:2) Z ( i )2 k i = 0 (cid:3) ≥ C ( δ ) e − δL . (18)Due to the local limit theorem, this gives on Γ( L, δ ), for large L so that e (1 − δ ) L ≥ e δL , (cid:88) n ≥ P ω [ Y n = 0] ≥ e (1 − δ ) L (cid:16) ce (1 − δ ) L/ (cid:17) m (cid:18) C ( δ ) e δL (cid:19) r ≥ c ( δ ) e [(1 − δ ) / − δr ] L , which goes to infinity as L goes to infinity due to our choice of δ , c ( δ ) being a posi-tive constant. Thanks to (17), this gives (cid:80) n ≥ P ω [ Y n = 0] = + ∞ for P -almost all ω .Consequently, due to (16), ( Y n ) n is recurrent for P -almost every environment ω . (cid:4) Remark 3.1.
Recall that Sinai [S82] (see also Golosov [G84] ) proved the convergence indistribution of (cid:0) Z ( i ) n / (log n ) (cid:1) n . Recall also that, due to de Moivre’s theorem, (cid:0) S ( i ) n / √ n (cid:1) n converges in distribution. Due to Theorem 1.1, Y is recurrent iff (cid:80) n / ( n m ((log n ) ) r ) = ∞ , where n m ((log n ) ) r is the product of the normalizations of the coordinates of Y underthe (non Markovian) annealed law P . Note also that Theorem 1.2 and the previous paragraph lead to the following statement(only for r ≥ (cid:80) n ≥ n m/ (log n ) I − < ∞ , then almost surely, S (1) n , . . . , S ( m ) n , Z (1) n , . . . Z ( r ) n meet simultaneously only a finite number of times; otherwise, they almost surely meetsimultaneously infinitely often.Now we will start to prove Theorem 1.2. Note that the case m ≤ Proof of Theorem 1.2.
Let A n := (cid:8) S (1) n = ... = S ( m ) n = Z (1) n = ... = Z ( r ) n (cid:9) for n ≥ Proof of (i).
Assume m = 3 and r = 1. Observe that for large n , P ω [ A n ] = (cid:88) k ∈ Z P ω (cid:2) Z (1) n = k (cid:3)(cid:0) P (cid:2) S (1) n = k (cid:3)(cid:1) ≤ C (cid:88) k ∈ Z P ω (cid:2) Z (1) n = k (cid:3) n √ n = Cn √ n for some C > k ∈ Z and n ∈ N , P (cid:2) S (1)2 n = k (cid:3) ≤ P (cid:2) S (1)2 n = 0 (cid:3) ∼ n → + ∞ ( πn ) − / due to the local limit theorem. Hence (cid:80) n P ω (cid:2) S (1) n = S (2) n = S (3) n = Z (1) n (cid:3) < ∞ and (i) follows by the Borel-Cantelli lemma in this case and a fortiori when m ≥ r ≥ Proof of (ii).
Assume m = 2 and r ≥ I = 1. Since I = 1, all the RWRE are in the sameenvironment, which is necessary to apply Proposition 1.5, which is essential to prove (ii).We use the generalization of the second Borel Cantelli lemma due to Kochen and Stone OLLISIONS OF SEVERAL WALKERS IN RECURRENT RANDOM ENVIRONMENTS 9 [KS64] combined with a result by Doob. To simplify notations, we also write ω for ω (1) ,so ω ( i ) = ω for every 1 ≤ i ≤ r .We first prove that (cid:80) n P ω [ A n ] = ∞ a.s. More precisely, we fix an initial condition x = ( x , x , y , ..., y r ) ∈ (2 Z ) r ∪ (2 Z + 1) r . We have for all n and ω , P xω [ A n ] = (cid:88) k ∈ Z P (cid:2) x + S (1) n = k (cid:3) P (cid:2) x + S (2) n = k (cid:3) r (cid:89) j =1 P y j ω (cid:2) Z ( j ) n = k (cid:3) . Notice that, for every i ∈ { , } , due to the de Moivre-Laplace theorem (see e.g. [LL10,Prop. 2.5.3 and Corollary 2.5.4],sup k ∈ ( x i + n +2 Z ) , | k |≤ (log n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P (cid:2) x i + S ( i ) n = k (cid:3) − √ √ πn e − ( k − x i ) / (2 n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = o ( n − / ) . Consequently for large even n , for every ω , P xω [ A n ] ≥ (cid:88) | k |≤ (log n ) ,k − ( x + n ) ∈ (2 Z ) πn r (cid:89) j =1 P y j ω (cid:2) Z ( j ) n = k (cid:3) = 1 πn (cid:88) | k |≤ (log n ) r (cid:89) j =1 P y j ω (cid:2) Z ( j ) n = k (cid:3) . This remains true for large odd n . Hence for large n , P xω [ A n ] ≥ πn P ( y ,...,y r ) ω (cid:2) Z (1) n = ... = Z ( r ) n (cid:3) − πn P y ω (cid:2)(cid:12)(cid:12) Z (1) n (cid:12)(cid:12) > (log n ) (cid:3) . (19)Recall that (cid:0) Z (1) n / (log n ) (cid:1) n converges almost surely to 0 with respect to the annealed law(see [DR86] Theorem 4, or more recently [HS98] Theorem 3). This holds also true for P y ω for P -almost every ω , so the last probability in (19) goes to 0 as n → + ∞ , which yieldslim N → + ∞ N (cid:80) Nn =1 1 n P y ω (cid:2)(cid:12)(cid:12) Z (1) n (cid:12)(cid:12) > (log n ) (cid:3) = 0. Hence for P -almost every ω ,lim sup N → + ∞ N N (cid:88) n =1 P xω [ A n ] ≥ c ( ω ) π , (20)with c ( ω ) := inf ( y ,...,y r ) ∈ [(2 Z ) r ∪ (2 Z +1) r ] lim sup N → + ∞ N (cid:80) Nn =1 1 n P ( y ,...,y r ) ω (cid:2) Z (1) n = ... = Z ( r ) n (cid:3) . If r = 1, then c ( ω ) = 1. If r >
1, due to Proposition 1.5, c ( ω ) > P -almost everyenvironment ω . This implies that (cid:88) n ≥ P xω [ A n ] = + ∞ . (21)Moreover, let C > n ≥ k ∈ Z , P (cid:2) S (1) n = k (cid:3) ≤ Cn − / , whichexists e.g. since P (cid:2) S (1)2 n = k (cid:3) ≤ P (cid:2) S (1)2 n = 0 (cid:3) ∼ n → + ∞ ( πn ) − / by the local limit theorem.So for 1 ≤ n < m , we have by Markov property, P xω [ A n ∩ A m ]= (cid:88) ( k,(cid:96) ) ∈ Z P ( y ,...,y r ) ω (cid:2) Z (1) n = ... = Z ( r ) n = k, Z (1) m = ... = Z ( r ) m = (cid:96) (cid:3) × P (cid:2) x + S (1) n = k (cid:3) P (cid:2) x + S (2) n = k (cid:3)(cid:0) P (cid:2) S (1) m − n = (cid:96) − k (cid:3)(cid:1) ≤ (cid:88) k ∈ Z P ( y ,...,y r ) ω (cid:2) Z (1) n = ... = Z ( r ) n = k (cid:3) P ( k,...,k ) ω (cid:2) Z (1) m − n = ... = Z ( r ) m − n (cid:3) C n ( m − n ) ≤ C n ( m − n ) . Consequently, for large N , (cid:88) ≤ n,m ≤ N,m (cid:54) = n P xω [ A n ∩ A m ] ≤ N (cid:88) n =1 C n N − n (cid:88) (cid:96) =1 (cid:96) ≤ C (log N ) . Applying this and (20) we get for P -almost every ω , for every initial condition x ∈ (2 Z ) r ∪ (2 Z + 1) r , lim sup N → + ∞ (cid:16)(cid:80) Nn =1 P xω [ A n ] (cid:17) (cid:80) ≤ n,m ≤ N P xω [ A n ∩ A m ] ≥ ( c ( ω )) π C . (22)Due to the Kochen and Stone extension of the second Borel-Cantelli lemma (see Item(iii) of the main theorem of [KS64] applied with X n = (cid:80) ni =1 A i , or [S76, p. 317]),(22) and (21) imply that P xω [ A n i.o. ] = P xω (cid:2) ∩ N ≥ ∪ n ≥ N A n (cid:3) ≥ ( c ( ω )) / (3 π C ) > P -almost every ω , due to a result by Doob(see for example Proposition V-2.4 in [N64]), since E := { A n i.o. } = ∩ N ≥ ∪ n ≥ N A n is invariant (with respect to the shifts of the sequence ( Y , Y , Y , . . . )), for every x ∈ (2 Z ) r ∪ (2 Z + 1) r , (cid:0) P ( S (1) n ,S (2) n ,Z (1) n ,...,Z ( r ) n ) ω [ E ] (cid:1) n converges P xω -almost surely to E . Butinf x ∈ (2 Z ) r ∪ (2 Z +1) r P xω [ E ] ≥ ( c ( ω )) / (3 π C ) >
0, so we conclude that E = 1 P xω -almostsurely, thus P xω ( E ) = 1, for P -almost every environment ω . Proof of (iii).
Assume m = 2 and r = I = 2. We have P [ A n ] = (cid:88) k ∈ Z P (cid:2) Z (1) n = Z (2) n = k (cid:3)(cid:0) P (cid:2) S (1) n = k (cid:3)(cid:1) ≤ C n P (cid:2) Z (1) n = Z (2) n (cid:3) = O (cid:0) n − (log n ) − / (cid:1) , due to Proposition 1.4 and the local limit theorem. Hence (cid:80) n P [ A n ] < ∞ and (iii)follows due to the Borel-Cantelli lemma. (cid:4) So there only remains to prove Propositions 1.4 and 1.5.4.
Probability of meeting for two independent recurrent rwre inindependent environments
The aim of this section is to prove Proposition 1.4, which is a key result in the proof ofcase (iii) of Theorem 1.2.Let Z (1) and Z (2) be two independent recurrent RWRE in independent environments ω (1) and ω (2) satisfying (4).The main idea of the proof is that Z (1) n and Z (2) n are localized with high (annealed) probabil-ity in two areas (depending on the environments, see Proposition 4.5) which have no com-mon point with high probability (see Lemma 4.6). Due to [S82], we know that, with highprobability, Z ( i ) n is close to the bottom B ( i ) n of some valley (containing 0 and of height largerthan log n ) for the potential V ( i ) . Here and in the following, V ( i ) denotes the potential OLLISIONS OF SEVERAL WALKERS IN RECURRENT RANDOM ENVIRONMENTS 11 corresponding to ω ( i ) , defined as in (5) with ω replaced by ω ( i ) . An intuitive idea to proveProposition 1.4 should then be that p n := P (cid:104) max i =1 , (cid:12)(cid:12) Z ( i ) n − B ( i ) n (cid:12)(cid:12) ≥ (cid:12)(cid:12) B (1) n − B (2) n (cid:12)(cid:12) / (cid:105) isvery small. More precisely we would like to prove that p n = O (cid:0) (log n ) − − ε (cid:1) . (In view ofthe proof of (iii) above, it would suffice to show that (cid:80) n p n n < ∞ ). However, this seemsdifficult to prove and we are not even sure that it is true. Indeed, in view of Lemma 4.4below (proved for a continuous approximation W ( i ) ≈ V ( i ) ), we think that with prob-ability greater than 1 / log n , 0 belongs to a valley of height between log n − n and log n and that the annealed probability that Z ( i ) n is close to the bottom of this valley(which is not B ( i ) n ) should be greater that 1 / log n . Hence, to prove Proposition 1.4, wewill work with several valleys instead of a single one.4.1. Proof of Proposition 1.4.
In this subsection, we use a Brownian motion W ( i ) ,approximating the potential V ( i ) , to build a localization domain Ξ n (cid:0) W ( i ) (cid:1) for Z ( i ) n , i ∈{ , } . This localization is stated in Proposition 4.5 and is crucial to prove Proposition1.4.In order to construct our localization domain Ξ n (cid:0) W ( i ) (cid:1) , we use the notion of h -extrema,defined as follows. Definition 4.1 ([NP89]) . If w : R → R is a continuous function and h > , we say that y ∈ R is an h -minimum for w if there exist real numbers a and c such that a < y < c , w ( y ) = inf [ a,c ] w , w ( a ) ≥ w ( y ) + h and w ( c ) ≥ w ( y ) + h . We say that y is an h -maximum for w if y is an h -minimum for − w . In any of these two cases, we say that y is an h -extremum for w . We also use the following notation.
Definition 4.2.
As in [C05] , we denote by W the set of functions w : R → R suchthat the three following conditions are satisfied: (a) w is continuous on R ; (b) for every h > , the set of h -extrema of w can be written { x k ( w, h ) , k ∈ Z } , with ( x k ( w, h )) k ∈ Z strictly increasing, unbounded from below and above, and with x ( w, h ) ≤ < x ( w, h ) ,notation that we use in the rest of the paper on W ; (c) for all k ∈ Z and h > , x k ( w, h ) is an h -minimum for w if and only if x k +1 ( w, h ) is an h -maximum for w . We now introduce, for w ∈ W , i ∈ Z and h > b i ( w, h ) := (cid:26) x i ( w, h ) if x ( w, h ) is an h -minimum ,x i +1 ( w, h ) otherwise . As a consequence, the b i ( w, h ) are the h -minima of w . We denote by M i ( w, h ) the unique h -maximum of w between b i ( w, h ) and b i +1 ( w, h ). That is, M i ( w, h ) = x j +1 ( w, h ) if b i ( w, h ) = x j ( w, h ).For w ∈ W , h > i ∈ Z , the restriction of w − w ( x i ( w, h )) to [ x i ( w, h ) , x i +1 ( w, h )]is denoted by T i ( w, h ) and is called an h -slope , as in [C05]. If x i ( w, h ) is an h -minimum(resp. h -maximum), then T i ( w, h ) is a nonnegative (resp. nonpositive) function, and itsmaximum (resp. minimum) is attained at x i +1 ( w, h ). We also introduce, for each slope T i ( w, h ), its height H ( T i ( w, h )) := | w ( x i +1 ( w, h )) − w ( x i ( w, h )) | ≥ h , and its excess height e ( T i ( w, h )) := H ( T i ( w, h )) − h ≥ When x i ( w, h ) is an h -minimum, the restriction of w to [ x i − ( w, h ) , x i +1 ( w, h )] will some-times be called valley of height at least h and of bottom x i ( w, h ). The height of this valleyis defined as min { w ( x i − ( w, h )) , w ( x i +1 ( w, h )) } − w ( x i ( w, h )), which can also be rewrittenmin { H ( T i − ( w, h )) , H ( T i ( w, h )) } .These h -extrema are useful to localize RWRE and diffusions in a random potential. In-deed, a diffusion in a two-sided Brownian potential W (resp. in a ( − κ/ W κ with 0 < κ <
1) is localized at large time t with high probability in a smallneighborhood of b ( W, log t ) (resp. some of the b i ( W κ , log t − √ log t ), i ≥
0) see e.g. [C05]and [C08] (resp. [AD15]). For some applications to recurrent RWRE, see e.g. [BF08] and[D14].Let C > α >
2. Define log (2) x = log log x for x >
1. As in [D14], we use theKoml´os-Major-Tusn´ady almost sure invariance principle [KMT75], which ensures that:
Lemma 4.3.
Up to an enlargement of (Ω , F , P ) , there exist two independent two-sidedBrownian motions (cid:0) W ( i ) ( s ) , s ∈ R (cid:1) ( i ∈ { , } ) with E (cid:2) ( W ( i ) (1)) (cid:3) = E [( V ( i ) (1)) ] = σ i and a real number ˜ C > such that for all n large enough, P (cid:34) sup | t |≤ (log n ) α (cid:12)(cid:12)(cid:12) V ( i ) ( (cid:98) t (cid:99) ) − W ( i ) ( t ) (cid:12)(cid:12)(cid:12) > ˜ C log (2) n (cid:35) ≤ (log n ) − C , i ∈ { , } . Proof.
Notice that V (1) and V (2) are independent, since ω (1) and ω (2) are independent. Dueto ([KMT75], Thm. 1), there exist positive constants a , b and c such that for N ∈ N , upto an enlargement of (Ω , F , P ), there exist two independent two-sided Brownian motions( W ( i ) ( s ) , s ∈ R ) ( i ∈ { , } ) on (Ω , F , P ) with E [( W ( i ) (1)) ] = E [( V ( i ) (1)) ] = σ i suchthat ∀ x ∈ R , ∀ i ∈ { , } , P (cid:34) sup | k |≤ N (cid:12)(cid:12)(cid:12) V ( i ) ( k ) − W ( i ) ( k ) (cid:12)(cid:12)(cid:12) > a log N + x (cid:35) ≤ be − cx . (23)Applying this result to N := (cid:98) (log n ) α (cid:99) + 1 and x := (log(2 b ) + C log (2) n ) /c and taking˜ C > (cid:0) aα + C c (cid:1) , we obtain that P (cid:34) sup | k |≤(cid:98) (log n ) α (cid:99) +1 (cid:12)(cid:12)(cid:12) V ( i ) ( k ) − W ( i ) ( k ) (cid:12)(cid:12)(cid:12) > ˜ C (2) n (cid:35) ≤
12 (log n ) − C , (24)for all n large enough. Moreover, for every n large enough, P (cid:34) sup | t |≤ (log n ) α (cid:12)(cid:12)(cid:12) W ( i ) ( (cid:98) t (cid:99) ) − W ( i ) ( t ) (cid:12)(cid:12)(cid:12) > ˜ C (2) n (cid:35) ≤ n ) α P (cid:34) sup ≤ t< (cid:12)(cid:12) W ( i ) ( t ) (cid:12)(cid:12) > ˜ C (2) n (cid:35) ≤ n ) α P (cid:34)(cid:12)(cid:12) W ( i ) (1) (cid:12)(cid:12) > ˜ C (2) n (cid:35) ≤ n ) α √ π e − ( ˜ C σ i (log (2) n ) = 12 √ π (log n ) α − ( ˜ C σ i log (2) n ≤
12 (log n ) − C , since sup [0 , W ( i ) = law (cid:12)(cid:12) W ( i ) (1) (cid:12)(cid:12) . This combined with (24) proves the lemma. (cid:4) OLLISIONS OF SEVERAL WALKERS IN RECURRENT RANDOM ENVIRONMENTS 13
In the rest of the paper, we use the W ( i ) introduced in Lemma 4.2. We will use the valleysfor the W ( i ) . Fix some C ≥ α + 2 + 10 ˜ C . Let h n := log n − C log (2) n . (25)We know from ([C05], Lemma 8) that W ( i ) ∈ W almost surely (recall definition 4.2).Moreover, using [HS98, Th 2.1] with 0 < a = b , we have P (cid:2) sup ≤ s ≤ t [ W ( i ) ( s ) − W ( i ) ( s )]
10 and b = h n , the following holds with a probability1 − o (cid:0) (log n ) − (cid:1) (since α > ∀ i ∈ { , } , − (log n ) α ≤ b − (cid:0) W ( i ) , h n (cid:1) ≤ M (cid:0) W ( i ) , h n (cid:1) ≤ (log n ) α . (26)The following lemma shows that Proposition 1.4 is more subtle than it may seem at firstsight. Lemma 4.4.
Let W be a two-sided standard Brownian motion and σ > . Then, forevery n large enough, P (cid:2) H ( T ( σW, h n )) ≤ log n (cid:3) ≥ C (log (2) n )(log n ) − , (27) P (cid:2) (cid:93) { j ∈ {− , ..., } , H ( T j ( σW, h n − C log (2) n )) ≤ log n + C log (2) n } ≥ (cid:3) (28)= O (cid:0) (log (2) n ) (log n ) − (cid:1) , P (cid:2) ∃ j ∈ {− , ..., } , H ( T j ( σW, h n − C log (2) n )) ≤ log n + C log (2) n (cid:3) (29)= O (cid:0) (log (2) n )(log n ) − (cid:1) . In particular, the probability that the height of the central valley for W ( i ) is less thanlog n is not negligible. However, with large enough probability, all the valleys close to 0except maybe one are large, with height greater than log n + C log (2) n . Proof of Lemma 4.4.
Let (cid:101) h n := h n − C log (2) n . First, due to ([NP89], Prop. 1, see also[C05] eq. (8)), e (cid:0) T i (cid:0) σW, (cid:101) h n (cid:1)(cid:1) / (cid:101) h n is for i (cid:54) = 0 an exponential random variable with mean1. Consequently, for i (cid:54) = 0 and large n , P (cid:2) H (cid:0) T i (cid:0) σW, (cid:101) h n (cid:1)(cid:1) ≤ log n + C log (2) n (cid:3) = P (cid:2) e (cid:0) T i (cid:0) σW, (cid:101) h n (cid:1)(cid:1) ≤ C log (2) n (cid:3) ≤ C log (2) n log n . Observe that e (cid:0) T (cid:0) σW, (cid:101) h n (cid:1)(cid:1) / (cid:101) h n is by scaling equal in law to e ( T ( W, x + 1) e − x (0 , ∞ ) ( x ) / n , C (log (2) n )(log n ) − ≤ P (cid:2) e (cid:0) T (cid:0) σW, (cid:101) h n (cid:1)(cid:1) ≤ C log (2) n (cid:3) ≤ P (cid:2) e (cid:0) T (cid:0) σW, (cid:101) h n (cid:1)(cid:1) ≤ C log (2) n (cid:3) ≤ C (log (2) n )(log n ) − . This remains true if (cid:101) h n is replaced by h n . These inequalities already prove (27) and (29).Moreover, due to ([NP89], Prop. 1), the slopes T i (cid:0) σW, (cid:101) h n (cid:1) , i ∈ Z are independent, up totheir sign, so the random variables H (cid:0) T i ( σW, (cid:101) h n ) (cid:1) , i ∈ Z are independent. This and theprevious inequalities lead to (28). (cid:4) Because of (27), it seems reasonable to consider strictly more than one valley of height atleast h n if we want to localize a recurrent RWRE with probability ≥ − (log n ) − ε for ε > i ∈ { , } , j ∈ Z and n ≥ n,j (cid:0) W ( i ) (cid:1) := (cid:8) x ∈ (cid:2) M j − (cid:0) W ( i ) , h n (cid:1) , M j (cid:0) W ( i ) , h n (cid:1)(cid:3) , W ( i ) ( x ) ≤ W ( i ) (cid:0) b j (cid:0) W ( i ) , h n (cid:1)(cid:1) + C log (2) n (cid:9) . Loosely speaking, Ξ n,j (cid:0) W ( i ) (cid:1) is the set of points with low potential in the j -th valley for W ( i ) . We also define Ξ n (cid:0) W ( i ) (cid:1) := (cid:91) j = − Ξ n,j (cid:0) W ( i ) (cid:1) . In Proposition 4.5 (proved in Section 4.2), we localize the RWRE Z ( i ) in a set of pointswhich are close to the b j ( . ) ”vertically”, instead of ”horizontally” as in Sinai’s theorem(see [S82]). Proposition 4.5.
Let ε > and i ∈ { , } . For all n large enough, we have P (cid:2) Z ( i ) n / ∈ Ξ n (cid:0) W ( i ) (cid:1)(cid:3) ≤ q n := (log n ) − ε . Proposition 1.4 is then an easy consequence of Proposition 4.5 and of the following esti-mate on the environments.
Lemma 4.6.
Let ε > . For large n , P (cid:2) Ξ n (cid:0) W (1) (cid:1) ∩ Ξ n (cid:0) W (2) (cid:1) (cid:54) = ∅ (cid:3) ≤ (log n ) − ε . Proof of Lemma 4.6.
First, let k ∈ Ξ n (cid:0) W ( i ) (cid:1) for some i ∈ { , } and n ≥
3. Hence k ∈ (cid:2) M j − (cid:0) W ( i ) , h n (cid:1) , M j (cid:0) W ( i ) , h n (cid:1)(cid:3) and W ( i ) ( k ) ≤ W ( i ) (cid:0) b j (cid:0) W ( i ) , h n (cid:1)(cid:1) + C log (2) n forsome j ∈ {− , − , , , } . By definition of h n -minima, we notice that the two Brownianmotions (cid:0) W ( i ) ( x + k ) − W ( i ) ( k ) , x ≥ (cid:1) and (cid:0) W ( i ) ( − x + k ) − W ( i ) ( k ) , x ≥ (cid:1) hit h n − C log (2) n before − C log (2) n . By independence, it follows that, for n large enough,for every k ∈ Z and i ∈ { , } , P (cid:2) k ∈ Ξ n (cid:0) W ( i ) (cid:1)(cid:3) ≤ P (cid:2) T +( i ) ( h n − C log (2) n ) < T +( i ) ( − C log (2) n ) (cid:3) × P (cid:2)(cid:12)(cid:12) T − ( i ) ( h n − C log (2) n ) (cid:12)(cid:12) < (cid:12)(cid:12) T − ( i ) ( − C log (2) n ) (cid:12)(cid:12)(cid:3) ≤ O (cid:16) ((log (2) n ) / log n ) (cid:17) , where T +( i ) ( z ) := inf { x > W ( i ) ( x ) = z } and T − ( i ) ( z ) := sup { x < W ( i ) ( x ) = z } .Consequently, since W (1) and W (2) are independent, we have uniformly on k ∈ Z , P (cid:2) k ∈ Ξ n (cid:0) W (1) (cid:1) ∩ Ξ n (cid:0) W (2) (cid:1)(cid:3) = P (cid:2) k ∈ Ξ n (cid:0) W (1) (cid:1)(cid:3) P (cid:2) k ∈ Ξ n (cid:0) W (2) (cid:1)(cid:3) ≤ O (cid:0) (log (2) n ) / (log n ) (cid:1) . (30)Finally, (26) applied with 2 + ε > α and (30) lead to P (cid:2) Ξ n (cid:0) W (1) (cid:1) ∩ Ξ n (cid:0) W (2) (cid:1) (cid:54) = ∅ (cid:3) OLLISIONS OF SEVERAL WALKERS IN RECURRENT RANDOM ENVIRONMENTS 15 ≤ o (cid:0) (log n ) − (cid:1) + (cid:98) (log n ) ε (cid:99) (cid:88) k = −(cid:98) (log n ) ε (cid:99) P (cid:2) k ∈ Ξ n (cid:0) W (1) (cid:1) ∩ Ξ n (cid:0) W (2) (cid:1)(cid:3) ≤ O (cid:16) (log (2) n ) (log n ) − ε (cid:17) ≤ (log n ) − ε , for every n large enough. Since this is true for every ε >
0, this proves the lemma. (cid:4)
Proof of Proposition 1.4.
We have for large n , due to Proposition 4.5, P (cid:2) Z (1) n = Z (2) n (cid:3) ≤ P (cid:2) Z (1) n = Z (2) n , Z (1) n ∈ Ξ n (cid:0) W (1) (cid:1) , Z (2) n ∈ Ξ n (cid:0) W (2) (cid:1)(cid:3) + P (cid:2) Z (1) n / ∈ Ξ n (cid:0) W (1) (cid:1)(cid:3) + P (cid:2) Z (2) n / ∈ Ξ n (cid:0) W (2) (cid:1)(cid:3) ≤ P (cid:2) Ξ n (cid:0) W (1) (cid:1) ∩ Ξ n (cid:0) W (2) (cid:1) (cid:54) = ∅ (cid:3) + 2 q n . This and Lemma 4.6 prove Proposition 1.4. (cid:4)
Proof of Proposition 4.5.
We fix i ∈ { , } . To simplify notations we write V for V ( i ) , Z n for Z ( i ) n and W for W ( i ) .The difficulty of this proof is that we have to localize Z n with probability 1 − (log n ) − ε instead of 1 − o (1) as Sinai did in [S82]. For this reason we need to take into accountsome cases which are usually considered to be negligible. In order to prove Proposition4.5, we first build a set G n of good environments, having high probability. We prove thaton such a good environment, the RWRE Z = ( Z n ) n will reach quickly the bottom b I ofone of the two valleys of W surrounding 0. We need to consider these two valleys becausewe cannot neglect the case in which 0 is close to the maximum of the potential betweenthese two valleys.Also, we cannot exclude that the valley surrounding b I is ”small”, that is, its heightis close to log n . Then, we have to consider two situations. If the height of this valley isquite larger than log n , then with large probability, Z stays in this valley up to time n (see Lemma 4.9). Otherwise (in the most difficult case, Lemma 4.11), Z can escape thevalley surrounding b I before time n , and in this case, with large probability, it reachesbefore time n the bottom b I of a neighbouring valley and stays in this valley up to time n . In both situations, we prove that Z n is localized in Ξ n ( W ), and more precisely in thedeepest places of the last valley visited before time n . In order to prove this localization,we use the invariant measure of a RWRE in our environment, started at b I or b I .We fix ε >
0. Recall (25). We introduce for j ∈ Z , x j := (cid:98) x j ( W, h n ) (cid:99) , b j := (cid:98) b j ( W, h n ) (cid:99) , M j := (cid:98) M j ( W, h n ) (cid:99) . We denote by G n the set of good environments ω satisfying (26) together with thefollowing properties (see Figure 1):sup | t |≤ (log n ) α (cid:12)(cid:12) V ( (cid:98) t (cid:99) ) − W ( t ) (cid:12)(cid:12) ≤ ˜ C log (2) n, (31) (cid:93) { j ∈ {− , ..., } , H ( T j ( W, h n − C log (2) n )) ≤ log n + C log (2) n } ≤ , (32) x − = M − x = b b x W ( k ) x = M − x = M x = b x = M h n h n C log nC log nx − = b − = M − k Ξ n, ( W ) C log n Ξ n, − ( W ) M +0 Ξ n, ( W ) h n h n + C log nA + − A −− Figure 1.
Pattern of W for a good environment ω ∈ G n and representation ofdifferent quantities. with h n defined in (25). For every n large enough, we have P (cid:2) ( G n ) c (cid:3) ≤ (log n ) − ε , (33)due to (26) and to Lemmas 4.3 and 4.4, since C > ω ∈ G + n where G + n := G n ∩ { x ( W, h n ) is an h n -minimum } , that is, b − ( W, h n ) = x − ( W, h n ) < x ( W, h n ) = M − ( W, h n ) ≤ < b ( W, h n ) = x ( W, h n ) . Indeed, the other case, that is, x ( W, h n ) is an h n -minimum, or equivalently ω ∈ G − n with G − n := G n \G + n , is similar by symmetry. Proof of Proposition 4.5.
Let us see how we can derive Proposition 4.5 from (33) and fromLemmas 4.7, 4.9 and 4.11 below. Applying Lemma 4.7 with y = 0 and j = − G + n , therandom walk Z goes quickly to b − or b with high probability. More precisely, setting E := { τ ( b − ) ∧ τ ( b ) ≤ n (log n ) − C } , there exists ˜ n ∈ N such that, for every n ≥ ˜ n , ∀ ω ∈ G + n , P ω ( E ) ≥ − (log n ) − . (34)Due to Lemmas 4.9 and 4.11, there exists ˜ n ∈ N such that, for every n ≥ ˜ n , ∀ ω ∈ G + n , P ω (cid:2) E , Z n / ∈ Ξ n ( W ) (cid:3) ≤ n ) − and so, using (34), ∀ n ≥ max(˜ n , ˜ n ) , ∀ ω ∈ G + n , P ω (cid:2) Z n / ∈ Ξ n ( W ) (cid:3) ≤ n ) − . By symmetry, this remains true with G + n replaced by G − n . Therefore, due to (33), for every n large enough, P (cid:2) Z n / ∈ Ξ n ( W ) (cid:3) ≤ (cid:90) G n P ω (cid:2) Z n / ∈ Ξ n ( W ) (cid:3) P (d ω ) + P (cid:2) ( G n ) c (cid:3) ≤ n ) − ε . Since this is true for every ε >
0, this proves Proposition 4.5. (cid:4)
OLLISIONS OF SEVERAL WALKERS IN RECURRENT RANDOM ENVIRONMENTS 17
We will use the following property. For j ∈ Z , let (cid:98) µ j ( x ) := exp (cid:2) − (cid:0) V ( x ) − V ( b j ) (cid:1)(cid:3) + exp (cid:2) − (cid:0) V ( x − − V ( b j ) (cid:1)(cid:3) = exp (cid:2) V ( b j ) (cid:3) µ ω ( x ) , x ∈ Z , with reversible measure µ ω defined in (14). It follows from reversibility that ∀ k ∈ N , ∀ x ∈ Z , ∀ b ∈ Z , P bω [ Z k = x ] = µ ω ( x ) µ ω ( b ) P xω [ Z k = b ] ≤ µ ω ( x ) µ ω ( b ) ≤ exp[ V ( b )] µ ω ( x ) . (35)In particular, ∀ j ∈ Z , ∀ k ∈ N , ∀ x ∈ Z , P b j ω [ Z k = x ] ≤ (cid:98) µ j ( x ) . (36) Lemma 4.7.
There exists n ∈ N such that, for every n ≥ n , every ω ∈ G n , every j ∈ {− , ..., } and every integer y ∈ ] b j , b j +1 [ , P yω (cid:2) τ ( b j ) ∧ τ ( b j +1 ) > n (log n ) − C (cid:3) ≤ (log n ) − . (37) Proof.
Let j ∈ {− , ..., } and ω ∈ G n . Assume for example that y ∈ [ M j , b j +1 [, the proofbeing symmetric in the case when y ∈ ] b j , M j ]. We set (see Figure 1 for j = − A + j := min( b j +1 , inf { k ≥ M j : W ( k ) ≤ W ( M j ( W, h n )) − h n − C log (2) n } ) ,A − j := max( b j , sup { k ≤ M j : W ( k ) ≤ W ( M j ( W, h n )) − h n − C log (2) n } ) .
1. If y ∈ [ M j , A + j [, due to (7), (26) and (31), applying Markov’s inequality, we get P yω (cid:34) τ ( A − j ) ∧ τ ( A + j ) > e h n +2 C log (2) n (cid:35) ≤ ε − [ b j +1 − b j ] e h n +2 C log (2) n exp (cid:104) max [ b j ,b j +1 ] V − min [ A − j ,A + j ] V (cid:105) ≤ ε − [ M − b − ] e h n +2 C log (2) n e (cid:0) W ( M j ( W,h n ))+ ˜ C log (2) n (cid:1) − (cid:0) W ( M j ( W,h n )) − h n − C log (2) n − ˜ C log (2) n − log ε − (cid:1) ≤ ε − (log n ) α e h n +( C +2 ˜ C ) log (2) n +log ε − e h n +2 C log (2) n ≤ ε − (log n ) α +2 ˜ C − C ≤
13 (log n ) − for every n large enough, where we used sup [ b j ( W,h n ) ,b j +1 ( W,h n )] W = W ( M j ( W, h n )) and V ( A ± j ) ≥ V ( A ± j ∓ − log − ε ε in the second line and C > α + 2 ˜ C + 2 in the last one.Hence by the strong Markov property, for n large enough, for every y ∈ [ M j , A + j ], P yω (cid:104) τ ( b j ) ∧ τ ( b j +1 ) > e h n +2 C log (2) n (cid:105) ≤
13 (log n ) − + P A − j ω (cid:104) τ ( b j ) > e h n +2 C log (2) n / (cid:105) + P A + j ω (cid:104) τ ( b j +1 ) > e h n +2 C log (2) n / (cid:105) . (38)2. Assume now that y ∈ [ A + j , b j +1 [ (and so A + j < b j +1 ). Observe that W admits no h n -maximum in the interval ] M j ( W, h n ) , b j +1 ( W, h n )] by definition of M j ( . ), somax M j ( W,h n ) ≤ u ≤ v ≤ b j +1 ( W,h n ) ( W ( v ) − W ( u )) < h n . Hence due to (7), (26), (31), and to Markov’s inequality, we have P yω (cid:104) τ ( M j ) ∧ τ ( b j +1 ) > e h n +2 C log (2) n / (cid:105) ≤ M − b − ] ε e h n +2 C log (2) n exp (cid:104) max M j ≤ (cid:96) ≤ k ≤ b j +1 (cid:0) V ( k ) − V ( (cid:96) ) (cid:1)(cid:105) ≤ ε − (log n ) α e h n +2 ˜ C log (2) n e h n +2 C log (2) n ≤
16 (log n ) − (39)for every n large enough, since 2 C > α + 2 ˜ C + 2. Moreover, due to (6), (26) and (31),and since there is no h n -maximum in [ A + j , b j +1 ] and so sup [ A + j ,b j +1 ] W < W ( A + j ) + h n , P yω [ τ ( M j ) < τ ( b j +1 )] ≤ (cid:18) b j +1 − (cid:88) (cid:96) = A + j e V ( (cid:96) ) (cid:19)(cid:18) b j +1 − (cid:88) (cid:96) = M j e V ( (cid:96) ) (cid:19) − ≤ (cid:2) b j +1 − A + j (cid:3) exp (cid:16) max (cid:96) ∈{ A + j ,...,b j +1 } V ( (cid:96) ) (cid:17) exp (cid:0) − V ( M j ) (cid:1) ≤ n ) α exp (cid:2) W ( A + j ) + h n − W ( M j ( W, h n )) + 2 ˜ C log (2) n (cid:3) ≤ n ) α − C +2 ˜ C ≤
16 (log n ) − (40)for every y ∈ [ A + j , b j +1 [ for all n large enough, since C > α + 2 ˜ C + 2. Gathering (39)and (40), we get, for all n large enough, for every y ∈ [ A + j , b j +1 [, uniformly on G n as theprevious inequalities, P yω (cid:2) τ ( b j +1 ) > n (log n ) − C / (cid:3) = P yω (cid:104) τ ( b j +1 ) > e h n +2 C log (2) n / (cid:105) ≤
13 (log n ) − , (41)recalling (25). This already proves (37) for y ∈ [ A + j , b j +1 [.3. For symmetry reasons, we also get that, for every n large enough, for every y ∈ ] b j , A − j ], P yω (cid:104) τ ( b j ) > e h n +2 C log (2) n / (cid:105) ≤
13 (log n ) − . (42)Finally, combining (38) with (41) and (42) proves (37) for y ∈ [ M j , A + j [. Hence, (37) istrue for y ∈ [ M j , b j +1 [ thanks to 2., and for y ∈ ] b j , M j ] by symmetry. This proves thelemma. (cid:4) We consider I ∈ {− , } such that τ (cid:0) b I (cid:1) = τ ( b − ) ∧ τ ( b ). Recall that E = E ( n ) := { τ (cid:0) b I (cid:1) ≤ α n } , where we set α n := n (log n ) − C . (43)We already saw in (34) that, thanks to Lemma 4.7 with y = 0 and j = −
1, we have ∀ ω ∈ G + n , P ω ( E ) ≥ − (log n ) − . We consider the event E = E ( n ) on which Z first goes to the bottom of a ”deep” valley: E +2 ( j ) := { W [ M j ( W, h n )] − W [ b j ( W, h n )] > log n + C log (2) n } , j ∈ Z ,E − ( j ) := { W [ M j − ( W, h n )] − W [ b j ( W, h n )] > log n + C log (2) n } , j ∈ Z ,E := E +2 ( I ) ∩ E − ( I ) . Notice that this event depends on ω but also on the first steps of Z up to time τ ( b I ).Similarly as in (29), this event happens with probability 1 − O ((log n ) − log (2) n ), so wecannot neglect E c . We will treat separately the two events E and E c (the study of E c OLLISIONS OF SEVERAL WALKERS IN RECURRENT RANDOM ENVIRONMENTS 19 being more complicated). Before considering these two events, we state the followinguseful result. We introduce for j ∈ Z , M + j := b j +1 ∧ inf { k ≥ M j , W ( k ) ≤ W ( b j ( W, h n )) + C log (2) n } , (44) M − j := b j − ∨ sup { k ≤ M j − , W ( k ) ≤ W ( b j ( W, h n )) + C log (2) n } , (45)where u ∨ v := max( u, v ), ( u, v ) ∈ R , so that ∀ k ∈ ] M − j , M j − ] ∪ [ M j , M + j [ , W ( k ) > W ( b j ( W, h n )) + C log (2) n. (46) Lemma 4.8.
For every n large enough, ∀ ω ∈ G n , ∀ j ∈ {− , . . . , } , sup k ≥ P b j ω (cid:0) Z k ∈ [ M − j , M + j ] \ Ξ n ( W ) (cid:1) ≤ (log n ) − . (47) Proof.
We claim that due to (31) and (26), V ( x ) ≥ W ( b j ( W, h n ))+ C log (2) n − ˜ C log (2) n − log ε − for every integer x ∈ (cid:0)(cid:2) M − j , M + j (cid:3) \ Ξ n,j ( W ) (cid:1) for j ∈ {− , . . . , } . This follows fromthe definition of Ξ n,j ( W ) if x ∈ [ M j − , M j ], and from (46) and the fact that | V ( y ) − V ( y − | ≤ log − ε ε otherwise. So, due to (36), (26) and (31), for large n , for all ω ∈ G n and j ∈ {− , . . . , } ,sup k ≥ P b j ω (cid:0) Z k ∈ [ M − j , M + j ] \ Ξ n ( W ) (cid:1) ≤ M + j (cid:88) x = M − j Ξ n ( W ) c ( x ) (cid:98) µ j ( x ) ≤ M + j (cid:88) x = M − j Ξ n,j ( W ) c ( x ) e V ( b j ) (cid:2) e − V ( x ) + e − V ( x )+log − ε ε (cid:3) ≤ n ) α ε − e (cid:0) W ( b j ( W,h n ))+ ˜ C log (2) n (cid:1) − (cid:0) W ( b j ( W,h n ))+( C − ˜ C ) log (2) n − log ε − (cid:1) = 2 ε − (log n ) α +2 ˜ C − C ≤ (log n ) − , since C ≥ α + 2 + 10 ˜ C . (cid:4) In the next lemma, we consider the case where Z goes quickly in a deep valley. Lemma 4.9 (Simplest case) . There exists n ∈ N such that for all n ≥ n , ∀ ω ∈ G + n , P ω ( E , E , Z n / ∈ Ξ n ( ω )) ≤ n ) − . Proof.
Due to (9), (26) and (31), we have for large n , for all ω ∈ G + n and all j ∈ {− , . . . , } uniformly on E +2 ( j ), P b j ω [ τ ( M j ) < n ] ≤ ne min [ bj,Mj − V − V ( M j − ≤ ne V ( b j ) − V ( M j − ≤ n exp (cid:2)(cid:0) W ( b j ( W, h n )) + ˜ C log (2) n (cid:1) − (cid:0) W ( M j ( W, h n )) − ˜ C log (2) n − log ε − (cid:1)(cid:3) ≤ ne − (log n + C log (2) n )+2 ˜ C log n +log ε − = ε − (log n ) C − C ≤ (log n ) − , (48)since C ≥ α + 2 + 10 ˜ C . Similarly, using (10) instead of (9), we have for large n , forall ω ∈ G + n and all j ∈ {− , . . . , } , uniformly on E − ( j ), P b j ω [ τ ( M j − ) < n ] ≤ (log n ) − . (49)Let τ ( x, y ) := inf { k ≥ , Z τ ( x )+ k = y } , x ∈ Z , y ∈ Z . In particular, on E ∩ E ∩ (cid:8) τ (cid:0) b I , M I − (cid:1) ≥ n (cid:9) ∩ (cid:8) τ (cid:0) b I , M I (cid:1) ≥ n (cid:9) , recalling (43), τ (cid:0) b I (cid:1) ≤ α n ≤ n ≤ τ (cid:0) b I (cid:1) + τ (cid:0) b I , M I − (cid:1) ∧ τ (cid:0) b I , M I (cid:1) , and so Z n ∈ (cid:2) M I − , M I (cid:3) ⊂ (cid:2) M −I , M + I (cid:3) . Applying (48) and (49) combined with thestrong Markov property at time τ ( b I ), and then (47), we get for large n , for every ω ∈ G + n , P ω [ E , E , Z n / ∈ Ξ n ( W )] ≤ P ω (cid:2) E , E , Z n / ∈ Ξ n ( W ) , τ (cid:0) b I , M I − (cid:1) ≥ n, τ (cid:0) b I , M I (cid:1) ≥ n (cid:3) + P ω (cid:2) E , τ (cid:0) b I , M I − (cid:1) < n (cid:3) + P ω (cid:2) E , τ (cid:0) b I , M I (cid:1) < n (cid:3) ≤ E ω (cid:2) E P b I ω (cid:0) Z n − k ∈ [ M I − , M I ] \ Ξ n ( W ) (cid:1) | k = τ ( b I ) (cid:3) + 2(log n ) − ≤ n ) − . (50)This proves the lemma. (cid:4) For the event E c , we will use the following lemma, which is actually true for any Markovchain. Lemma 4.10.
Let a (cid:54) = b . We have, ∀ k ∈ N , P bω [ τ ( a ) = k ] ≤ P bω [ τ ( a ) < τ ( b )] . Proof.
Let k ∈ N ∗ . We have, by the Markov property, P bω [ τ ( a ) = k ] = k (cid:88) n =0 P bω (cid:2) τ ( a ) = k, Z n = b, ∀ n < (cid:96) ≤ k, Z (cid:96) (cid:54) = b (cid:3) ≤ k (cid:88) n =0 P bω (cid:2) Z n = b (cid:3) P bω (cid:2) τ ( a ) = k − n, τ ( a ) < τ ( b ) (cid:3) ≤ P bω [ τ ( a ) ∈ [0 , k ] , τ ( a ) < τ ( b )] ≤ P bω [ τ ( a ) < τ ( b )] , where we used P bω [ Z n = b ] ≤ (cid:4) Lemma 4.11 (Most difficult case) . There exists n (cid:48) ∈ N such that for all n ≥ n (cid:48) , ∀ ω ∈ G + n , P ω ( E , E c , Z n / ∈ Ξ n ( ω )) ≤ n ) − . Proof.
An essential remark is that if we are on E c with ω ∈ G + n , then, due to (32), eitherwe are on E − ( I ) \ E +2 ( I ) or on E +2 ( I ) \ E − ( I ). In the first case we set I := I + 1 , A := M + I , B := M I ( W, h n ) and D := M −I . whereas in the second case we set I := I − , A := M −I , B := M I − ( W, h n ) and D := M + I . Loosely speaking, with large probability, b I is the bottom of the second valley reachedby Z , and Z can reach it before time n or not, so we have to consider both cases.We introduce τ (cid:48) ( A, b I ) := inf { k ≥ , Z τ ( b I )+ τ ( b I ,A )+ k = b I } and E := { τ ( b I ) + τ ( b I , A ) < n − n (log n ) − C } ∩ { τ (cid:48) ( A, b I ) ≤ n (log n ) − C } , OLLISIONS OF SEVERAL WALKERS IN RECURRENT RANDOM ENVIRONMENTS 21 E := { τ ( b I ) + τ ( b I , A ) ∈ [ n − n (log n ) − C , n ] } ,E := { τ ( b I ) + τ ( b I , A ) > n } ∩ { τ ( b I , D ) > n } ,E := { τ ( b I , D ) ≤ n } ,E := { τ (cid:48) ( A, b I ) ≥ n (log n ) − C } . Notice that E c ⊂ E ∪ E ∪ E ∪ E ∪ E . (51) • Control on E . First, E c ∩ {I = I + 1 } ⊂ E − ( I ), so by (49) and since D = M −I < M I − < b I when I = I + 1, we have for large n for every ω ∈ G + n , P ω ( E c ∩ {I = I + 1 } ∩ E ) ≤ E ω (cid:2) E − ( I ) P b I ω (cid:0) τ ( M I − ) < τ ( D ) ≤ n (cid:1)(cid:3) ≤ (log n ) − . (52)The case I = I − P ω ( E c ∩ E ) ≤ n ) − for large n for every ω ∈ G + n . • Control on E . We start by proving that for every n large enough, for every ω ∈ G + n , uniformly on E c , ∀ x ∈ N , P b I ω (cid:2) τ ( A ) ∈ [ n − n (log n ) − C − x, n − x ] (cid:3) ≤ (log n ) − . (53)Using Lemma 4.10 and then (6), we obtain on E − ( I ) \ E +2 ( I ), since b I < M I log n + C log (2) n . For the same reason, there is no ( h n − C log (2) n )-extrema between B and b I ( W, h n ), and so sup B ≤ u ≤ v ≤ b I ( W,h n ) ( W ( v ) − W ( u )) < h n − C log (2) n inthe case I = I + 1. Hence in this case, due to (7), (26), (31) and to Markov’sinequality, and since (cid:98) B (cid:99) = M I < A < b I , P Aω (cid:2) τ ( (cid:98) B (cid:99) ) ∧ τ ( b I ) ≥ n (log n ) − C (cid:3) ≤ ε − n ) α e h n − C log (2) n +2 ˜ C log (2) n n (log n ) − C ≤ ε − n ) α − C +2 ˜ C ≤
12 (log n ) − , (57)for every n large enough since C ≥ α + 2 + 10 ˜ C . This is also true in the case I = I − I = I + 1, we have max [ A,b I ] V ≤ sup [ M + I ,b I ( W,h n )] W + ˜ C log (2) n ≤ W ( M + I ) + ( h n − C log (2) n ) + ˜ C log (2) n due tothe previous remark, (26) and (31). Also, W ( M + I ) ≤ W ( b I ( W, h n )) + C log (2) n by (44), otherwise we would have M + I = b I +1 and W ( b I +1 ) ≥ W ( b I ( W, h n )) + C log (2) n ≥ W ( M I ( W, h n )) − log n due to our hypothesis in this case I = I + 1,which in turn would give W ( M I ( W, h n )) − W ( b I +1 ( W, h n )) ≤ log n + 2 ˜ C log (2) n ,which contradicts (32) since 2 ˜ C < C . So by (6), (26) and (31), recalling that W ( b I ( W, h n )) + h n ≤ W ( M I ( W, h n )) and B = M I , we get P Aω [ τ ( (cid:98) B (cid:99) ) < τ ( b I )] ≤ (cid:0) b I − A (cid:1) exp (cid:104) max [ A,b I ] V − V ( B ) (cid:105) ≤ n ) α e (cid:0) W ( b I ( W,h n ))+ h n +( ˜ C − C ) log (2) n (cid:1) − (cid:0) W ( M I ( W,h n )) − ˜ C log (2) n (cid:1) ≤ n ) α +2 ˜ C − C ≤ (log n ) − / n large enough since C ≥ α + 2 + 10 ˜ C . We prove similarly (58) in thecase I = I −
1. Then, (57) and (58) prove (56). Finally, (56) combined with thestrong Markov property lead to (55). • Control on E . On E ∩ E c ∩ E , we have τ ( b I ) ≤ n ≤ τ ( b I )+ τ ( b I , A ) ∧ τ ( b I , D )and in particular Z n ∈ [ A ∧ D, A ∨ D ] = [ M −I , M + I ]. Applying (47) as in thesimplest case, we get for large n , for all ω ∈ G n , P ω ( E ∩ E c ∩ E , Z n / ∈ Ξ n ( W )) ≤ E ω (cid:2) { τ ( b I ) ≤ α n } P b I ω (cid:0) Z n − k ∈ [ M −I , M + I ] \ Ξ n ( W ) (cid:1) | k = τ ( b I ) (cid:3) ≤ (log n ) − . (59) • Control on E . On E ∩ E c ∩ E , we have τ ( b I ) ≤ τ ( b I )+ τ ( b I , A )+ τ (cid:48) ( A, b I ) < n .Moreover, the height of the valley [ M I − ( W, h n ) , M I ( W, h n )] is at least log n + C log (2) n on E c due to (32), that is, we are on E − ( I ) ∩ E +2 ( I ). Also we get P b I ω (cid:2) τ (cid:0) M I − (cid:1) ∧ τ (cid:0) M I (cid:1) < n (cid:3) ≤ n ) − by (48) and (49) uniformly on E c ∩ G + n for large n . Using (47) and [ M I − , M I ] ⊂ [ M −I , M + I ], this gives for large n forevery ω ∈ G + n , P ω ( E ∩ E c ∩ E , Z n / ∈ Ξ n ( W )) ≤ E ω (cid:2) { τ ( b I ) This section is devoted to the proof of Proposition 1.5, which is a consequence of thefollowing proposition whose proof is deferred.Let r > Z (1) , ..., Z ( r ) be r independent recurrent RWRE in the same environment ω satisfying (4). Proposition 5.1. Let δ ∈ (0 , . There exist events ∆ N ( δ ) , N ≥ and (cid:98) b ( N ) ∈ Z depending only on the environment ω , and constants c ( δ ) > , ε ( δ ) ∈ (0 , , with lim inf N → + ∞ P (cid:2) ∆ N ( δ ) (cid:3) ≥ − δ, (61) such that ∀ ( y , . . . , y r ) ∈ (2 Z ) r , ∃ N ∈ N , ∀ N ≥ N , ∀ ω ∈ ∆ N ( δ ) , ∀ j ∈ { , . . . , r } , ∀ n ∈ (cid:2) N − ε ( δ ) , N (cid:3) ∩ (2 N ) , P y j ω (cid:2) Z ( j ) n = (cid:98) b ( N ) (cid:3) ≥ c ( δ ) . (62) This remains true if (2 Z ) r and N are replaced respectively by (2 Z + 1) r and N + 1 .Proof of Proposition 1.5. Let δ ∈ (0 , P (cid:2) lim sup N → + ∞ ∆ N ( δ ) (cid:3) = P (cid:34) (cid:92) N ∈ N (cid:91) n ≥ N ∆ n ( δ ) (cid:35) = lim N → + ∞ P (cid:20) (cid:91) n ≥ N ∆ n ( δ ) (cid:21) ≥ lim inf N → + ∞ P [∆ N ( δ )] ≥ − δ. (63)Now, let ( y , . . . , y r ) ∈ (2 Z ) r . There exists N ∈ N such that for every N ≥ N , on ∆ N ( δ ), N (cid:88) n =1 n (cid:88) k ∈ Z r (cid:89) j =1 P y j ω (cid:2) Z ( j ) n = k (cid:3) ≥ N (cid:88) n = N − ε ( δ ) N ( n ) n [ c ( δ )] r ≥ [ c ( δ )] r ε ( δ )4 log N if N is large enough. Consequently, we have on lim sup N → + ∞ ∆ N ( δ ) = { ω ∈ ∆ N ( δ ) i.o. } ,lim sup N → + ∞ N N (cid:88) n =1 n (cid:88) k ∈ Z r (cid:89) j =1 P y j ω (cid:2) Z ( j ) n = k (cid:3) ≥ [ c ( δ )] r ε ( δ )4 > . This and (63) prove Proposition 1.5 in the case ( y , . . . , y r ) ∈ (2 Z ) r . The proof in thecase ( y , . . . , y r ) ∈ (2 Z + 1) r is similar. (cid:4) Now, it remains to prove Proposition 5.1. Main idea of the proof of Proposition 5.1. Let Z be a RWRE as in Section 2.In order to prove that Z n is localized at (cid:98) b ( N ) with a quenched probability P y j ω greaterthan a positive constant, we use a coupling argument between a copy of Z starting from (cid:98) b ( N ) and a RWRE (cid:98) Z reflected in some valley around (cid:98) b ( N ), under its invariant probabilitymeasure. To this aim, we approximate the potential V by a Brownian motion W , use W to build the set of good environments ∆ N ( δ ) and estimate its probability P (cid:2) ∆ N ( δ ) (cid:3) , andthen define (cid:98) b ( N ).We build ∆ N as the intersection of 7 events ∆ ( i ) N , i = 0 , . . . , 6. First, ∆ (0) N gives anapproximation of V by W . Loosely speaking ∆ (1) N guarantees that the central valley(containing the origin) of height log n has a height much larger than log n , so that Z willnot escape this valley before time n (see Lemma 5.6). ∆ (1) N also ensures that this centralvalley does not contain sub-valleys of height close to log n , so that with high quenchedprobability, Z reaches quickly the bottom of this valley without being trapped in suchsubvalleys (see Lemma 5.5). To this aim, we also need that the bottom of this valley isnot too far from 0, which is given by ∆ (3) N , and that the value of the potential between 0and the bottom of this valley is low enough, which is given by ∆ (4) N and ∆ (2) N . Additionally,∆ (5) N is useful to provide estimates for the invariant probability measure (cid:98) ν , and is useful toprove that the coupling occurs quickly (Lemma 5.9, using Lemmas 5.7 and 5.8). Finally,∆ (6) N says that (cid:98) ν (cid:0)(cid:98) b ( N ) (cid:1) , which is roughly the invariant probability measure at the bottomof the central valley, is larger than a positive constant.5.2. Construction of ∆ N ( δ ) . Let δ ∈ (0 , N ( δ ) satisfying (61) and (62), and the proof of (61). Wewill construct ∆ N ( δ ) as an intersection∆ N ( δ ) := (cid:92) i =0 ∆ ( i ) N , (64)where the sets ∆ ( i ) N , defined below, also depend on δ . In what follows, ε i is for i > 0a positive constant depending on δ and used to define the set ∆ ( i ) N . As in the previoussection, we will approximate the potential V by a two-sided Brownian motion W suchthat Var ( W (1)) = Var ( V (1)) (see Figure 2 for patterns of the potential V and of W in∆ N ( δ )). We start with ∆ (1) N , . . . , ∆ (5) N which are W -measurable. Using the same notationas before for h -extrema, for a two-sided Brownian motion W , we define∆ (1) N := { W ∈ W} ∩ (cid:92) i = − (cid:8) H [ T i ( W, (1 − ε ) log N )] ≥ (1 + 2 ε ) log N (cid:9) , (65)∆ ( R ) N := (cid:8) x (cid:0) W, (1 − ε ) log N (cid:1) is a ((1 − ε ) log N )-minimum for W (cid:9) , (66)and ∆ ( L ) N := (cid:2) ∆ ( R ) N (cid:3) c , where R stands for right and L for left, ∆ (2) N := ∆ (2 ,R ) N ∪ ∆ (2 ,L ) N with∆ (2 ,R ) N := (cid:26) max (cid:2) ,x (cid:0) W, (1 − ε ) log N (cid:1)(cid:3) W < W (cid:0) x (cid:0) W, (1 − ε ) log N (cid:1)(cid:1) − ε log N (cid:27) ∩ ∆ ( R ) N , (67) OLLISIONS OF SEVERAL WALKERS IN RECURRENT RANDOM ENVIRONMENTS 25 ∆ (2 ,L ) N := (cid:26) max (cid:2) x (cid:0) W, (1 − ε ) log N (cid:1) , (cid:3) W < W (cid:0) x (cid:0) W, (1 − ε ) log N (cid:1)(cid:1) − ε log N (cid:27) ∩ ∆ ( L ) N ; (68)∆ (3) N := (cid:110) − ε − (log N ) ≤ x − [ W, (1 − ε ) log N ] ≤ x [ W, (1 − ε ) log N ] ≤ ε − (log N ) (cid:111) ;(69)∆ (4) N := ∩ i =0 {| W ( x i ( W, (1 − ε ) log N )) | > ε log N } (70)and ∆ (5) N := ∆ (5 ,R ) N ∪ ∆ (5 ,L ) N , where∆ (5 ,L ) N := (cid:26) min (cid:2) ,x (cid:0) W, (1 − ε ) log N (cid:1)(cid:3) W > W (cid:0) x (cid:0) W, (1 − ε ) log N (cid:1)(cid:1) + ε log N (cid:27) ∩ ∆ ( L ) N , (71)∆ (5 ,R ) N := (cid:26) min (cid:2) x (cid:0) W, (1 − ε ) log N (cid:1) , (cid:3) W > W (cid:0) x (cid:0) W, (1 − ε ) log N (cid:1)(cid:1) + ε log N (cid:27) ∩ ∆ ( R ) N . (72) Lemma 5.2. Let W be a two-sided Brownian motion such that Var ( W (1)) = Var ( V (1)) .There exist ( ε , ε , ε , ε , ε ) ∈ (0 , / with ε = ε such that, for every i ∈ { , ..., } , P [∆ ( i ) N ] > − δ/ . x x = b x ≤ ε − (log N ) b x ε log Nx − ≥ − ε − (log N ) ε log N WV ( k ) β ( R ) N ≈ b b ( N ) (1 + ε ) log Nθ ( R ) N ≥ ε log N (1 − ε ) log N (1 − ε ) log N b L + b L − ≥ (1 + 2 ε ) log N ≥ (1 + 2 ε ) log N Figure 2. Pattern of the potential V and of W for ω ∈ ∆ N ∩ ∆ ( R ) N , where x i denotes x i ( W, (1 − ε ) log N ). Proof. First, by the same arguments as in the proof of Lemma 4.4, there exists ε ∈ (0 , / 10) such that P (cid:2) ∆ (1) N (cid:3) ≥ − δ/ We now introduce (cid:102) W N ( x ) := W ( x (log N ) ) / log N , which has the same law as W byscaling. We notice that x (cid:0)(cid:102) W N , − ε (cid:1) is a local extremum for (cid:102) W N , so P (cid:2) x (cid:0)(cid:102) W N , − ε (cid:1) = 0 (cid:3) = 0. Hence we have x (cid:0)(cid:102) W N , − ε (cid:1) < < x (cid:0)(cid:102) W N , − ε (cid:1) a.s. Westart with the case where x (cid:0)(cid:102) W N , − ε (cid:1) is a (1 − ε )-minimum for (cid:102) W N , that is, thebottom b ( W, (1 − ε ) log N ) of the central valley of depth at least (1 − ε ) log N for W is on the right. That is, we assume we are on ∆ ( R ) N ∩ W . Since (cid:102) W N is continuous on (cid:2) , x (cid:0)(cid:102) W N , − ε (cid:1)(cid:3) , (cid:102) W N attains its maximum on this interval at some y ∈ (cid:2) , x (cid:0)(cid:102) W N , − ε (cid:1)(cid:3) . So, (cid:102) W N ( y ) ∈ (cid:2) , (cid:102) W N (cid:0) x (cid:0)(cid:102) W N , − ε (cid:1)(cid:1)(cid:3) , since max [ x ( (cid:102) W N , − ε ) ,x ( (cid:102) W N , − ε )] (cid:102) W N = (cid:102) W N (cid:0) x (cid:0)(cid:102) W N , − ε (cid:1)(cid:1) . If (cid:102) W N ( y ) = (cid:102) W N (cid:0) x (cid:0)(cid:102) W N , − ε (cid:1)(cid:1) , then y would be a (1 − ε )-maximum for (cid:102) W N , with x (cid:0)(cid:102) W N , − ε (cid:1) < y < x (cid:0)(cid:102) W N , − ε (cid:1) , which is not possible on W . So, (cid:102) W N ( y ) = max (cid:2) ,x (cid:0) (cid:102) W N , − ε (cid:1)(cid:3) (cid:102) W N < (cid:102) W N (cid:0) x (cid:0)(cid:102) W N , − ε (cid:1)(cid:1) . Consequently, thereexists ε ∈ (0 , / 10) such that P max (cid:2) ,x (cid:0) (cid:102) W N , − ε (cid:1)(cid:3) (cid:102) W N < (cid:102) W N (cid:0) x (cid:0)(cid:102) W N , − ε (cid:1)(cid:1) − ε (cid:12)(cid:12)(cid:12)(cid:12) ∆ ( R ) N ≥ − δ , and the same is true if we exchange x and x by symmetry (and then [0 , x ( . . . )] isreplaced by [ x ( . . . ) , ( R ) N by ∆ ( L ) N ). Hence P (cid:2) ∆ (2) N (cid:3) ≥ − δ/ 10 by scaling.Moreover, there exists ε ∈ (0 , / 10) such that P (cid:2) ∆ (3) N (cid:3) = P (cid:2) − ε − ≤ x − [ (cid:102) W N , − ε ] ≤ x [ (cid:102) W N , − ε ] ≤ ε − (cid:3) ≥ − δ/ 10, where we get the first equality by scaling.Finally, there exists ε ∈ (0 , / 10) such that P (cid:2) ∆ (4) N (cid:3) ≥ − δ/ 10, by scaling, since (cid:12)(cid:12)(cid:102) W N (cid:0) x i (cid:0)(cid:102) W N , − ε (cid:1)(cid:1)(cid:12)(cid:12) > i ∈ { , } . Indeed, x (cid:0)(cid:102) W N , − ε (cid:1) < 0, so (cid:102) W N (cid:0) x (cid:0)(cid:102) W N , − ε (cid:1)(cid:1) = max [ x ( (cid:102) W N , − ε ) , (cid:102) W N > ( R ) N ∩W , and (cid:102) W N (cid:0) x (cid:0)(cid:102) W N , − ε (cid:1)(cid:1) = min [ x ( (cid:102) W N , − ε ) , (cid:102) W N < ( L ) N ∩ W , so (cid:12)(cid:12)(cid:102) W N (cid:0) x (cid:0)(cid:102) W N , − ε (cid:1)(cid:1)(cid:12)(cid:12) > (cid:12)(cid:12)(cid:102) W N (cid:0) x (cid:0)(cid:102) W N , − ε (cid:1)(cid:1)(cid:12)(cid:12) > W by − W in ∆ (2) N proves that with ε := ε > 0, the event ∆ (5) N satisfies P (cid:2) ∆ (5) N (cid:3) = P (cid:2) ∆ (2) N (cid:3) ≥ − δ/ (cid:4) From now on, ε , ..., ε are the ones given by Lemma 5.2. Let ε := min( ε , . . . , ε ) / . (73) Lemma 5.3. Up to an enlargement of (Ω , F , P ) , there exist a two-sided Brownian motion ( W ( s ) , s ∈ R ) defined on Ω such that Var ( W (1)) = Var ( V (1)) and a real number ξ > such that P (cid:34) sup | t |≤ ε − (log N ) (cid:12)(cid:12) V ( (cid:98) t (cid:99) ) − W ( t ) (cid:12)(cid:12) > ε log N (cid:35) = O ( N − ξ ) . Proof. Due to (23) (applied with N replaced by 2 ε − (log N ) and x = ( ε/ 2) log N − a log[2 ε − (log N ) ]), there exists for N large enough, possibly on an enlarged probability OLLISIONS OF SEVERAL WALKERS IN RECURRENT RANDOM ENVIRONMENTS 27 space, a Brownian motion ( W ( s ) , s ∈ R ) such that P (cid:34) sup | k |≤ ε − (log N ) | V ( k ) − W ( k ) | > ε N (cid:35) ≤ N − c ε and such that Var ( W (1)) = Var ( V (1)). Moreover, P (cid:34) sup | t |≤ ε − (log N ) (cid:12)(cid:12) W ( t ) − W ( (cid:98) t (cid:99) ) (cid:12)(cid:12) > ε N (cid:35) ≤ ε − (log N ) P (cid:34) sup | t |≤ | W ( t ) | > ε N (cid:35) = O ((log N ) exp[ − ε (log N ) / (8 σ )]) . Combining these two inequalities proves the lemma. (cid:4) Recall (64): it remains to define ∆ (0) N and ∆ (6) N , see (74) and (76) below, and then we claim Lemma 5.4. For large N , P (cid:2) ∆ N ( δ ) (cid:3) ≥ − δ . Hence (61) holds true.Proof. From now on, W is the Brownian motion W coming from Lemma 5.3 and ∆ (1) N , ..., ∆ (5) N are the corresponding events defined in (65)–(72). We set∆ (0) N := (cid:40) sup | t |≤ ε − (log N ) (cid:12)(cid:12) V ( (cid:98) t (cid:99) ) − W ( t ) (cid:12)(cid:12) ≤ ε log N (cid:41) . (74)For N large enough, P (cid:2) ∆ (0) N (cid:3) > − δ/ 10 by Lemma 5.3. In particular on the event ∆ (0) N ∩ ∆ (3) N , we can apply the inequalities of ∆ (0) N to any t ∈ (cid:2) x − (cid:0) W, (1 − ε ) log N (cid:1) , x (cid:0) W, (1 − ε ) log N (cid:1)(cid:3) , since those t satisfy | t | ≤ ε − (log N ) . We now introduce (here this is for V directly, not for W ) θ ( R ) N := inf (cid:110) i ∈ N , V ( i ) − min ≤ j ≤ i V ( j ) ≥ (1 + ε ) log N (cid:111) ,β ( R ) N := sup (cid:110) i < θ ( R ) N , V ( i ) = min ≤ j ≤ θ ( R ) N V ( j ) (cid:111) ,θ ( L ) N := sup (cid:110) i ∈ ( − N ) , V ( i ) − min i ≤ j ≤ V ( j ) ≥ (1 + ε ) log N (cid:111) ,β ( L ) N := inf (cid:110) i > θ ( L ) N , V ( i ) = min θ ( L ) N ≤ j ≤ V ( j ) (cid:111) . By ([DGPS07], eq. (4.33)), there exists ε > N is large enough, P (cid:2) ∆ (6 ,R ) N (cid:3) ≥ − δ/ 10, where∆ (6 ,R ) N := (cid:40) θ ( R ) N − (cid:88) i =0 e − (cid:2) V ( i ) − V (cid:0) β ( R ) N (cid:1)(cid:3) ≤ ε − (cid:41) , ∆ (6 ,L ) N := (cid:40) − (cid:88) i = θ ( L ) N e − (cid:2) V ( i ) − V (cid:0) β ( L ) N (cid:1)(cid:3) ≤ ε − (cid:41) . (75)Replacing V ( . ) by V ( − . ) gives P (cid:2) ∆ (6 ,L ) N (cid:3) ≥ − δ/ 10. Consequently, P (cid:2) ∆ (6) N (cid:3) ≥ − δ/ (6) N := ∆ (6 ,R ) N ∩ ∆ (6 ,L ) N . (76)This, combined with Lemma 5.2 and P (cid:2) ∆ (0) N (cid:3) > − δ/ 10, proves the lemma. (cid:4) Random walk in an environment ω ∈ ∆ N ( δ ) . The aim of this subsection is toprove Proposition 5.1 with the ∆ N ( δ ) constructed in the previous section, see (64)–(72),(74) and (76). Let δ ∈ (0 , N for ∆ N ( δ ). We also fix ( y , . . . , y r ) ∈ (2 Z ) r .There exists N ∈ N such that for N ≥ N , P (cid:2) ∆ N (cid:3) ≥ − δ (due to Lemma 5.4), a ≤ ε log N , and max ≤ j ≤ r | y j | < min( ε , ε )(log N ) / (4 a ), where we set a := log((1 − ε ) /ε ).We introduce, recalling (66), (cid:98) b ( N ) := 2 (cid:4) β ( R ) N / (cid:5) ∆ ( R ) N + 2 (cid:4) β ( L ) N / (cid:5) ∆ ( L ) N . (77)We will carry out the proof in the case ω ∈ ∆ N ∩ ∆ ( R ) N . The case ω ∈ ∆ N ∩ ∆ ( L ) N is similarby symmetry. We define (cid:98) x i := (cid:98) x i ( W, (1 − ε ) log N ) (cid:99) , and D (1) N := (cid:8) τ (cid:0)(cid:98) b ( N ) (cid:1) < τ ( (cid:98) x ) (cid:9) , D (2) N := (cid:8) τ ( (cid:98) x ) ∧ τ (cid:0)(cid:98) b ( N ) (cid:1) ≤ N − ε (cid:9) . We sometimes write x i instead of x i ( W, (1 − ε ) log N ) in the following.In the following lemma, we prove that Z goes quickly to (cid:98) b ( N ), which is nearly the bottomof the potential V in the central valley (cid:2)(cid:98) x , (cid:98) x (cid:3) , with large probability under P y j ω , uniformlyon ∆ N ∩ ∆ ( R ) N and j . Lemma 5.5. There exists N ∈ N such that for all N ≥ N , ∀ ω ∈ ∆ N ∩ ∆ ( R ) N , ∀ j ∈ { , . . . , r } , P y j ω (cid:2) D (1) N (cid:3) ≥ − N − ( ε ∧ ε ) / , P y j ω (cid:2) D (2) N (cid:3) ≥ − N − ε / . Proof. Let N ≥ N , ω ∈ ∆ N ∩ ∆ ( R ) N and j ∈ { , . . . , r } . First, notice that W ( x ) − W ( x ) = H [ T ( W, (1 − ε ) log N )] ≥ (1 + 2 ε ) log N because ω ∈ ∆ (1) N . This gives, recalling (73) V ( (cid:98) x ) − V ( (cid:98) x ) ≥ W ( x ) − W ( x ) − ε log N ≥ (1 + ε ) log N (78)since ω ∈ ∆ (3) N ∩ ∆ (0) N (see (74) and the remark after it). Hence 0 ≤ (cid:98) b ( N ) ≤ β ( R ) N ≤ θ ( R ) N ≤ (cid:98) x ≤ ε − (log N ) .Now, assume that θ ( R ) N < x . Since V (cid:0) θ ( R ) N (cid:1) − V (cid:0) β ( R ) N (cid:1) ≥ (1 + ε ) log N , the previousinequalities would give, on ∆ (0) N ∩ ∆ (3) N , W (cid:0) θ ( R ) N (cid:1) − W (cid:0) β ( R ) N (cid:1) ≥ (1 + ε − ε ) log N ≥ (1 − ε ) log N . So, recalling that W ( x ) = min [0 ,x ] W , there would exist a ((1 − ε ) log N )-maximum for W in ]0 , x [, which is not possible. Hence x ≤ θ ( R ) N .So, V (cid:0) β ( R ) N (cid:1) ≤ V ( (cid:98) x ) ≤ W ( x ) + ε log N < − ε (log N ) / ω ∈ ∆ (0) N ∩ ∆ (3) N ∩ ∆ (4) N .If y j > 0, then min [0 ,y j ] V ≥ −| y j | a ≥ − ε (log N ) / > V ( β ( R ) N ) + 2 a , because N ≥ N .Since similarly, max [0 ,y j ] V ≤ ε (log N ) / ε < 1, we get successively y j ≤ θ ( R ) N and y j ≤ β ( R ) N − ≤ (cid:98) b ( N ) − 1. If y j < 0, we prove similarly that (cid:98) x < y j since V ( (cid:98) x ) ≥ ε (log N ) / 9. Hence in every case, (cid:98) x < y j < (cid:98) b ( N ).We now prove that max [ y j , (cid:98) b ( N )] V − V ( (cid:98) x ) ≤ − [( ε ∧ ε ) / 2] log N. (79)To this aim, notice that max [0 , (cid:98) x ] V − V ( (cid:98) x ) ≤ − ε (log N ) / ω ∈ ∆ (2 ,R ) N ∩ ∆ (0) N ∩ ∆ (3) N ,and that if y j < 0, we have max [ y j , V − V ( (cid:98) x ) ≤ | y j | a − (8 / ε log N ≤ − ε (log N ) / W ( x ) ≥ ε log N on ∆ (2 ,R ) N and so V ( (cid:98) x ) ≥ (8 / ε log N . This gives (79) when (cid:98) b ( N ) ≤ (cid:98) x . OLLISIONS OF SEVERAL WALKERS IN RECURRENT RANDOM ENVIRONMENTS 29 Assume now (cid:98) x < (cid:98) b ( N ). We have seen after (78) that 0 ≤ (cid:98) b ( N ) ≤ θ ( R ) N ≤ (cid:98) x , moreover, V (cid:0)(cid:98) b ( N ) (cid:1) ≤ V (cid:0) β ( R ) N (cid:1) + a and we have proved that V (cid:0) β ( R ) N (cid:1) ≤ V ( (cid:98) x ), so we obtain V (cid:0)(cid:98) b ( N ) (cid:1) − ε log N − a ≤ V (ˆ x ) − ε log N ≤ W ( x ) ≤ W (cid:0)(cid:98) b ( N ) (cid:1) ≤ V (cid:0)(cid:98) b ( N ) (cid:1) + ε log N since W ( x ) = min [ x ,x ] W and ω ∈ ∆ (3) N ∩ ∆ (0) N , so that (cid:12)(cid:12) W ( x ) − V (cid:0)(cid:98) b ( N ) (cid:1)(cid:12)(cid:12) ≤ ε log N + a ≤ ε log N ≤ ε , ε )(log N ) / . (80)Moreover there is no ((1 − ε ) log N )-maximum for W in ( x , x ), therefore,max [ x , (cid:98) b ( N )] W < W (cid:0)(cid:98) b ( N ) (cid:1) + (1 − ε ) log N ≤ W ( x ) + (1 − ε + 3 ε / 9) log N , (81)by ∆ (0) N applied to (cid:98) b ( N ) followed by (80). Since V ( (cid:98) x ) ≥ V ( (cid:98) x ) + (1 + ε ) log N on∆ (1) N ∩ ∆ (0) N ∩ ∆ (3) N , this gives max [ (cid:98) x , (cid:98) b ( N )] V − V ( (cid:98) x ) ≤ − ε log N (since ω ∈ ∆ (0) N ∩ ∆ (3) N ).Recapitulating all this gives (79) also when (cid:98) x < (cid:98) b ( N ).So by (6) and (79), we get uniformly on ∆ N ∩ ∆ R ) N and j for large N , P y j ω (cid:2)(cid:0) D (1) N (cid:1) c (cid:3) ≤ (cid:2)(cid:98) b ( N ) − y j (cid:3) exp (cid:104) max [ y j , (cid:98) b ( N )] V − V ( (cid:98) x ) (cid:105) ≤ ε − (log N ) N ( ε ∧ ε ) / ≤ N − ( ε ∧ ε ) / , where we used ω ∈ ∆ (3) N and (cid:98) x < y j < (cid:98) b ( N ) < (cid:98) x . This proves the first inequality of thelemma.We now turn to D (2) N . Notice that (cid:12)(cid:12)(cid:98) b ( N ) − (cid:98) x (cid:12)(cid:12) ≤ | (cid:98) x − (cid:98) x | ≤ ε − (log N ) on ∆ N since 0 ≤ (cid:98) b ( N ) ≤ (cid:98) x as proved after (78). Moreover, there is no ((1 − ε ) log N )-maximum for W in ( x , x ), so max x ≤ u ≤ v ≤ x ( W ( v ) − W ( u )) < (1 − ε ) log N . Also if x < (cid:98) b ( N ), min [ x , (cid:98) b ( N )] W = W ( x ) and (81) lead to max x ≤ u ≤ v ≤ (cid:98) b ( N ) ( W ( v ) − W ( u )) ≤ (1 − ε + 3 ε / 9) log N . Since ω ∈ ∆ (0) N ∩ ∆ (3) N , this givesmax (cid:98) x ≤ (cid:96) ≤ k ≤ (cid:98) b ( N ) (cid:0) V ( k ) − V ( (cid:96) ) (cid:1) ≤ (1 − ε / 9) log N. (82)Hence, we have by (7), E y j ω (cid:2) τ ( (cid:98) x ) ∧ τ (cid:0)(cid:98) b ( N ) (cid:1)(cid:3) ≤ [ (cid:98) b ( N ) − (cid:98) x ] ε exp (cid:104) max (cid:98) x ≤ (cid:96) ≤ k ≤ (cid:98) b ( N ) (cid:0) V ( k ) − V ( (cid:96) ) (cid:1)(cid:105) ≤ N ) N − ε ε ε . So due to Markov’s inequality, P y j ω (cid:2)(cid:0) D (2) N (cid:1) c (cid:3) ≤ N − ε / , uniformly in ω ∈ ∆ N ∩ ∆ ( R ) N and j , for large N . (cid:4) In the following lemma, we prove that with large quenched probability, uniformly on∆ N ∩ ∆ ( R ) N , after first hitting (cid:98) b ( N ), the random walk Z stays in the central valley (cid:2)(cid:98) x , (cid:98) x (cid:3) at least up to time N . To this aim, we now define D (3) N := (cid:8) ∀ k ∈ (cid:2) τ (cid:0)(cid:98) b ( N ) (cid:1) , τ (cid:0)(cid:98) b ( N ) (cid:1) + N − (cid:3) , (cid:98) x < Z k < (cid:98) x (cid:9) . Lemma 5.6. We have for large N , ∀ ω ∈ ∆ N ∩ ∆ ( R ) N , ∀ j ∈ { , . . . , r } , P y j ω (cid:2) D (3) N (cid:3) = P (cid:98) b ( N ) ω (cid:2) τ (cid:0)(cid:98) x (cid:1) ∧ τ (cid:0)(cid:98) x (cid:1) ≥ N (cid:3) ≥ − e a N − ε . Proof. Let ω ∈ ∆ N ∩ ∆ ( R ) N . We recall that | V ( k ) − V ( k − | ≤ a for every k ∈ Z . Wehave, since x ≤ θ ( R ) N and so V ( β ( R ) N ) ≤ V ( (cid:98) x ), and by (78), V (cid:0)(cid:98) b ( N ) (cid:1) − V ( (cid:98) x ) ≤ V (cid:0)(cid:98) x (cid:1) + a − V ( (cid:98) x ) ≤ a − (1 + ε ) log N. (83)Similarly, V (cid:0)(cid:98) b ( N ) (cid:1) − V ( (cid:98) x ) ≤ a − (1 + ε ) log N. (84)Hence (9) and (10) lead respectively to P (cid:98) b ( N ) ω ( τ ( (cid:98) x ) < N ) ≤ N exp (cid:16) min [ (cid:98) b ( N ) , (cid:98) x − V − V ( (cid:98) x − (cid:17) ≤ N e a − (1+ ε ) log N ≤ e a N − ε ,P (cid:98) b ( N ) ω ( τ ( (cid:98) x ) < N ) ≤ N exp (cid:16) min [ (cid:98) x , (cid:98) b ( N ) − V − V ( (cid:98) x ) (cid:17) ≤ N e a − (1+ ε ) log N ≤ e a N − ε . These two inequalities yield P (cid:98) b ( N ) ω (cid:2) τ (cid:0)(cid:98) x (cid:1) ∧ τ (cid:0)(cid:98) x (cid:1) < N (cid:3) ≤ e a N − ε , uniformly on ∆ N ∩ ∆ ( R ) N , which proves the lemma. (cid:4) Now, similarly as in Brox [B86] for diffusions in random potentials (see also [AD15, p.45]), we introduce a coupling between Z (under P (cid:98) b ( N ) ω ) and a reflected random walk (cid:98) Z defined below. More precisely, we define, for fixed N , (cid:98) ω (cid:98) x := 1, (cid:98) ω x := ω x if (cid:98) x < x < (cid:98) x ,and (cid:98) ω x := 0. We consider a random walk (cid:0) (cid:98) Z n (cid:1) n in the environment (cid:98) ω , starting from x ∈ (cid:2)(cid:98) x , (cid:98) x (cid:3) , and denote its law by P x (cid:98) ω . That is, (cid:98) Z satisfies (2) with (cid:98) ω instead of ω and ω ( j ) and (cid:98) Z instead of Z ( j ) . In words, (cid:98) Z is a random walk in the environment ω , startingfrom x ∈ (cid:2)(cid:98) x , (cid:98) x (cid:3) , and reflected at (cid:98) x and (cid:98) x . Also, let (cid:98) µ ( (cid:98) x ) := e − V ( (cid:98) x ) , (cid:98) µ ( (cid:98) x ) := e − V ( (cid:98) x − , (cid:98) µ ( x ) := e − V ( x ) + e − V ( x − , (cid:98) x < x < (cid:98) x , and (cid:98) µ ( x ) = 0 if x / ∈ [ (cid:98) x , (cid:98) x ]. Notice that (cid:98) µ ( . ) / (cid:98) µ ( Z ) is an invariant probability measure for (cid:98) Z . As a consequence, (cid:98) ν ( x ) := (cid:98) µ ( x ) Z ( x ) / (cid:98) µ (2 Z ) , x ∈ Z , (85)is an invariant probability measure for (cid:0) (cid:98) Z n (cid:1) n for fixed (cid:98) ω . That is, P (cid:98) ν (cid:98) ω (cid:0) (cid:98) Z k = x (cid:1) = (cid:98) ν ( x )for every x ∈ Z and k ∈ N , where P (cid:98) ν (cid:98) ω ( . ) := (cid:80) x ∈ Z (cid:98) ν ( x ) P x (cid:98) ω ( . ). Notice that (cid:98) ν and (cid:98) µ dependon N and ω .We can now, again for fixed N and ω , build a coupling Q ω of Z and (cid:98) Z , such that Q ω (cid:0) (cid:98) Z ∈ . (cid:1) = P (cid:98) ν (cid:98) ω (cid:0) (cid:98) Z ∈ . (cid:1) , Q ω ( Z ∈ . ) = P (cid:98) b ( N ) ω ( Z ∈ . ) , (86)such that under Q ω , these two Markov chains move independently until τ (cid:98) Z = Z := inf (cid:8) k ≥ , (cid:98) Z k = Z k (cid:9) , which is their first meeting time, then (cid:98) Z k = Z k for every τ (cid:98) Z = Z ≤ k < τ exit , where τ exit isthe next exit time of Z from the central valley [ (cid:98) x , (cid:98) x ], that is, τ exit := inf (cid:8) k > τ (cid:98) Z = Z , Z k / ∈ [ (cid:98) x , (cid:98) x ] (cid:9) , and then (cid:98) Z and Z move independently again after τ exit . OLLISIONS OF SEVERAL WALKERS IN RECURRENT RANDOM ENVIRONMENTS 31 Now, we would like to prove that under Q ω , Z and (cid:98) Z collide quickly, that is, τ (cid:98) Z = Z is verysmall compared to N . To this aim, we introduce (cid:98) L − := sup { k ≤ (cid:98) b ( N ) , V ( k ) − V (cid:0)(cid:98) b ( N ) (cid:1) ≥ (1 − ε ) log N } , (cid:98) L + := inf { k ≥ (cid:98) b ( N ) , V ( k ) − V (cid:0)(cid:98) b ( N ) (cid:1) ≥ (1 − ε ) log N } . Let u ∨ v := max( u, v ). We have the following: Lemma 5.7. We have for large N , τ ( . ) denoting hitting times by Z as before, ∀ ω ∈ ∆ N ∩ ∆ ( R ) N , Q ω (cid:2) τ (cid:0)(cid:98) L − (cid:1) ∨ τ (cid:0)(cid:98) L + (cid:1) > N − ε / (cid:3) ≤ N − ε / . Proof. Let N ≥ N and ω ∈ ∆ N ∩ ∆ ( R ) N . Notice that (cid:98) x ≤ (cid:98) L − < (cid:98) b ( N ) < (cid:98) L + ≤ θ ( R ) N ≤ (cid:98) x similarly as after (78). Because ω ∈ ∆ (5 ,R ) N ∩ ∆ (0) N ∩ ∆ (3) N and due to (80), we have since ε = ε , ∀ k ∈ [ (cid:98) x , − , V ( k ) − V (cid:0)(cid:98) b ( N ) (cid:1) ≥ W ( x )+( ε − ε ) log N − V (cid:0)(cid:98) b ( N ) (cid:1) ≥ ( ε / 2) log N. (87)Moreover, recalling a = log((1 − ε ) /ε ), we have min [0 ,θ ( R ) N ] V = V (cid:0) β ( R ) N (cid:1) ≥ V (cid:0)(cid:98) b ( N ) (cid:1) − a ,so min [ (cid:98) x , (cid:98) L + ] V ≥ min [ (cid:98) x ,θ ( R ) N ] V ≥ V (cid:0)(cid:98) b ( N ) (cid:1) − a . Notice also for further use that, for every k ∈ (cid:2) θ ( R ) N , (cid:98) x (cid:3) , we have V (cid:0) θ ( R ) N (cid:1) − V ( k ) ≤ W (cid:0) θ ( R ) N (cid:1) − W ( k ) + 2 ε log N < (1 − ε + 2 ε ) log N since ω ∈ ∆ (0) N ∩ ∆ (3) N and because there is no ((1 − ε ) log N )–maximum for W in( (cid:98) x , (cid:98) x (cid:1) and (cid:98) x ≤ θ ( R ) N ≤ k ≤ (cid:98) x , as proved after (78). Since V (cid:0) θ ( R ) N (cid:1) − V (cid:0)(cid:98) b ( N ) (cid:1) ≥ (1 + ε ) log N − a , this gives ∀ k ∈ (cid:2) θ ( R ) N , (cid:98) x (cid:3) , V ( k ) − V (cid:0)(cid:98) b ( N ) (cid:1) = V ( k ) − V (cid:0) θ ( R ) N (cid:1) + V (cid:0) θ ( R ) N (cid:1) − V (cid:0)(cid:98) b ( N ) (cid:1) ≥ ε log N. (88)Putting together these inequalities gives in particular min [ (cid:98) x , (cid:98) x ] V ≥ V (cid:0)(cid:98) b ( N ) (cid:1) − a . Fur-thermore, max [ (cid:98) b ( N ) , (cid:98) L + ] V ≤ V (cid:0)(cid:98) b ( N ) (cid:1) + (1 − ε ) log N + a . (89)Hence, max (cid:98) x ≤ (cid:96) ≤ k ≤ (cid:98) L + − , k ≥ (cid:98) b ( N ) [ V ( k ) − V ( (cid:96) )] ≤ max [ (cid:98) b ( N ) , (cid:98) L + ] V − min [ (cid:98) x , (cid:98) L + ] V ≤ (1 − ε ) log N + 2 a . This, (7), Markov’s inequality and ω ∈ ∆ (3) N give P (cid:98) b ( N ) ω (cid:2) τ ( (cid:98) x ) ∧ τ (cid:0)(cid:98) L + (cid:1) > N − ε / (cid:3) ≤ N − (1 − ε / ε − ε − (log N ) N − ε e a ≤ N − ε / uniformly for large N . Moreover by (6), (84), (89) and since ω ∈ ∆ (3) N , P (cid:98) b ( N ) ω (cid:2) τ ( (cid:98) x ) < τ (cid:0)(cid:98) L + (cid:1)(cid:3) ≤ (cid:0)(cid:98) L + − (cid:98) b ( N ) (cid:1) exp (cid:2) max [ (cid:98) b ( N ) , (cid:98) L + ] V − V (cid:0)(cid:98) x (cid:1)(cid:3) ≤ (log N ) e a ε N ε ≤ N ε / uniformly for large N . Consequently, Q ω (cid:2) τ (cid:0)(cid:98) L + (cid:1) > N − ε / (cid:3) = P (cid:98) b ( N ) ω (cid:2) τ (cid:0)(cid:98) L + (cid:1) > N − ε / (cid:3) ≤ P (cid:98) b ( N ) ω (cid:2) τ ( (cid:98) x ) ∧ τ (cid:0)(cid:98) L + (cid:1) > N − ε / (cid:3) + P (cid:98) b ( N ) ω (cid:2) τ ( (cid:98) x ) < τ (cid:0)(cid:98) L + (cid:1)(cid:3) ≤ N − ε / . We prove similarly that Q ω (cid:2) τ (cid:0)(cid:98) L − (cid:1) > N − ε / (cid:3) ≤ N − ε / uniformly for large N , using(8) and (83) instead of (7) and (84) respectively, and because min [ (cid:98) x , (cid:98) x ] V ≥ V (cid:0)(cid:98) b ( N ) (cid:1) − a which we proved after (88). This proves Lemma 5.7. (cid:4) Lemma 5.8. For large N , ∀ ω ∈ ∆ N ∩ ∆ ( R ) N , (cid:98) ν (cid:0)(cid:2)(cid:98) x , (cid:98) L − (cid:3)(cid:1) + (cid:98) ν (cid:0)(cid:2)(cid:98) L + , (cid:98) x (cid:3)(cid:1) ≤ N − ε / . Proof. Let N ≥ N and ω ∈ ∆ N ∩ ∆ ( R ) N . Recall that (cid:98) x ≤ (cid:98) L − < (cid:98) b ( N ) < (cid:98) L + ≤ (cid:98) x ,which is proved before (87). Notice that (cid:98) L − ≤ x ≤ (cid:98) L + , which is proved similarly as x ≤ θ ( R ) N after (78). Using the same method as for (88) with (cid:98) L + instead of θ ( R ) N , we get V ≥ V (cid:0)(cid:98) b ( N ) (cid:1) + ( ε / 3) log N on (cid:2)(cid:98) L + , (cid:98) x (cid:3) . Also, V (cid:0)(cid:98) L + − (cid:1) ≥ V (cid:0)(cid:98) b ( N ) (cid:1) + ( ε / 3) log N Since (cid:98) µ (2 Z ) ≥ e − V ( (cid:98) b ( N )) , this leads to (cid:98) ν (cid:0)(cid:2)(cid:98) L + , (cid:98) x (cid:3)(cid:1) ≤ (cid:2)(cid:98) x − (cid:98) L + + 2 (cid:3) e − V ( (cid:98) b ( N )) N − ε / / (cid:98) µ (2 Z ) ≤ ε − (log N ) N − ε / ≤ N − ε / / N , where we used ω ∈ ∆ (3) N . We prove similarly that (cid:98) ν (cid:0)(cid:2)(cid:98) x , (cid:98) L − (cid:3)(cid:1) ≤ N − ε / / N , which ends the proof of the lemma. (cid:4) Lemma 5.9. There exists N ∈ N such that for N ≥ N for every ω ∈ ∆ N ∩ ∆ ( R ) N , Q ω (cid:2) τ (cid:98) Z = Z > N − ε / (cid:3) ≤ N − ε / (90) and Q ω [ τ exit ≤ N ] ≤ Q ω (cid:2) τ ( (cid:98) x ) ∧ τ ( (cid:98) x ) < N (cid:3) = P (cid:98) b ( N ) ω (cid:2) τ ( (cid:98) x ) ∧ τ ( (cid:98) x ) < N (cid:3) ≤ e a N − ε . (91) Proof. Due to Lemma 5.7, we have for large N for all ω ∈ ∆ N ∩ ∆ ( R ) N , Q ω (cid:2) τ (cid:98) Z = Z > N − ε / (cid:3) ≤ Q ω (cid:2) τ (cid:0)(cid:98) L − (cid:1) ∨ τ (cid:0)(cid:98) L + (cid:1) < τ (cid:98) Z = Z (cid:3) + Q ω (cid:2) τ (cid:0)(cid:98) L − (cid:1) ∨ τ (cid:0)(cid:98) L + (cid:1) > N − ε / (cid:3) ≤ Q ω (cid:2) τ (cid:0)(cid:98) L − (cid:1) < τ (cid:98) Z = Z , (cid:98) Z < (cid:98) b ( N ) (cid:3) + Q ω (cid:2) τ (cid:0)(cid:98) L + (cid:1) < τ (cid:98) Z = Z , (cid:98) Z ≥ (cid:98) b ( N ) (cid:3) + 4 N − ε / . Notice that under Q ω , Z = (cid:98) b ( N ) ∈ (2 Z ) by (86) and (77), and (cid:98) Z ∈ (2 Z ) by (86) and(85). So the process (cid:0) (cid:98) Z k − Z k (cid:1) k ∈ N starts at (cid:0) (cid:98) Z − (cid:98) b ( N ) (cid:1) ∈ (2 Z ) and only makes jumpsbelonging to {− , , } , and thus up to time τ (cid:98) Z = Z − < > 0) on (cid:8) (cid:98) Z < (cid:98) b ( N ) (cid:9)(cid:0) resp. (cid:8) (cid:98) Z ≥ (cid:98) b ( N ) (cid:9)(cid:1) , and in particular at time τ (cid:0)(cid:98) L − (cid:1) on (cid:8) τ (cid:0)(cid:98) L − (cid:1) < τ (cid:98) Z = Z , (cid:98) Z < (cid:98) b ( N ) (cid:9)(cid:0) resp. at time τ (cid:0)(cid:98) L + (cid:1) on (cid:8) τ (cid:0)(cid:98) L + (cid:1) < τ (cid:98) Z = Z , (cid:98) Z ≥ (cid:98) b ( N ) (cid:9)(cid:1) . This gives for large N for all ω ∈ ∆ N ∩ ∆ ( R ) N , Q ω (cid:2) τ (cid:98) Z = Z > N − ε / (cid:3) ≤ Q ω (cid:2) τ (cid:0)(cid:98) L − (cid:1) < τ (cid:98) Z = Z , (cid:98) Z τ ( (cid:98) L − ) < (cid:98) L − (cid:3) + Q ω (cid:2) τ (cid:0)(cid:98) L + (cid:1) < τ (cid:98) Z = Z , (cid:98) Z τ ( (cid:98) L + ) > (cid:98) L + (cid:3) + 4 N − ε / ≤ Q ω (cid:2) τ (cid:0)(cid:98) L − (cid:1) < τ (cid:98) Z = Z , (cid:98) Z (cid:98) τ ( (cid:98) L − ) / (cid:99) ≤ (cid:98) L − (cid:3) + Q ω (cid:2) τ (cid:0)(cid:98) L + (cid:1) < τ (cid:98) Z = Z , (cid:98) Z (cid:98) τ ( (cid:98) L + ) / (cid:99) ≥ (cid:98) L + (cid:3) +4 N − ε / ≤ (cid:98) ν (cid:0)(cid:2)(cid:98) x , (cid:98) L − (cid:3)(cid:1) + (cid:98) ν (cid:0)(cid:2)(cid:98) L + , (cid:98) x (cid:3)(cid:1) + 4 N − ε / , where the last inequality comes from the fact that Q ω (cid:0) (cid:98) Z k = x (cid:1) = P (cid:98) ν (cid:98) ω (cid:0) (cid:98) Z k = x (cid:1) = (cid:98) ν ( x )for every x ∈ Z and every (deterministic) k ∈ N as explained after (85), and from the OLLISIONS OF SEVERAL WALKERS IN RECURRENT RANDOM ENVIRONMENTS 33 independence of (cid:98) Z with Z and then τ ( . ) up to τ (cid:98) Z = Z . Now, applying Lemma 5.8, this gives(90) for large N for every ω ∈ ∆ N ∩ ∆ ( R ) N .Due to (86) and Lemma 5.6, for large N for every ω ∈ ∆ N ∩ ∆ ( R ) N , (91) holds. (cid:4) Proof of Proposition 5.1. Recall that we have fixed δ ∈ (0 , 1) and that (61) comes fromLemma 5.4. Let us prove (62). To this aim, we fix ( y , . . . , y r ) ∈ (2 Z ) r . Let N ∈ N besuch that N ≥ max( N , N , N ) and such that for every N ≥ N , ε − (log N ) [ N − ε / +2 N − ε ] ≤ ε − , N − ( ε ∧ ε ∧ ε ) / ≤ / N − ε / ≥ N − ε + N − ε / and 5 N − ε / +2 e a N − ε ≤ ε e − a / 6, recalling a = log((1 − ε ) /ε ). Now, we would like to give a lower bound for P y j ω (cid:2) Z n = (cid:98) b ( N ) (cid:3) for n even. Recall (64) and (66). Let N ≥ N , ω ∈ ∆ N ∩ ∆ ( R ) N , j ∈ { , . . . , r } , and n ∈ (2 N ), with n ∈ [ N − ε + N − ε / , N ]. We have by the strongMarkov property, P y j ω (cid:2) Z n = (cid:98) b ( N ) (cid:3) ≥ P y j ω (cid:2) Z n = (cid:98) b ( N ) , τ (cid:0)(cid:98) b ( N ) (cid:1) ≤ N − ε (cid:3) = E y j ω (cid:2) { τ ( (cid:98) b ( N )) ≤ N − ε } P (cid:98) b ( N ) ω (cid:0) Z k = (cid:98) b ( N ) (cid:1) | k = n − τ ( (cid:98) b ( N )) (cid:3) ≥ P y j ω (cid:2) τ (cid:0)(cid:98) b ( N ) (cid:1) ≤ N − ε (cid:3) inf k ∈ [ N − ε / ,N ] ∩ (2 N ) P (cid:98) b ( N ) ω (cid:0) Z k = (cid:98) b ( N ) (cid:1) ≥ (cid:0) − N − ( ε ∧ ε ) / − N − ε / (cid:1) inf k ∈ [ N − ε / ,N ] ∩ (2 N ) P (cid:98) b ( N ) ω (cid:0) Z k = (cid:98) b ( N ) (cid:1) (92)because (cid:98) b ( N ) and y j are even (see (77)) and then τ (cid:0)(cid:98) b ( N ) (cid:1) is also even under P y j ω , andwhere we used Lemma 5.5 in the last line. Moreover, for k ∈ [ N − ε / , N ] ∩ (2 N ), P (cid:98) b ( N ) ω (cid:0) Z k = (cid:98) b ( N ) (cid:1) = Q ω (cid:0) Z k = (cid:98) b ( N ) (cid:1) ≥ Q ω (cid:0) Z k = (cid:98) b ( N ) , τ (cid:98) Z = Z ≤ N − ε / , τ exit > N (cid:1) = Q ω (cid:0) (cid:98) Z k = (cid:98) b ( N ) , τ (cid:98) Z = Z ≤ N − ε / , τ exit > N (cid:1) ≥ Q ω (cid:0) (cid:98) Z k = (cid:98) b ( N ) (cid:1) − Q ω (cid:0) τ (cid:98) Z = Z > N − ε / (cid:1) − Q ω (cid:0) τ exit ≤ N (cid:1) ≥ (cid:98) ν (cid:0)(cid:98) b ( N ) (cid:1) − N − ε / − e a N − ε , (93)where we used (86) in the first and last line, Z k = (cid:98) Z k for k ∈ (cid:2) τ (cid:98) Z = Z , τ exit (cid:1) under Q ω inthe third line, and Q ω (cid:0) (cid:98) Z k = x (cid:1) = P (cid:98) ν (cid:98) ω (cid:0) (cid:98) Z k = x (cid:1) = (cid:98) ν ( x ) since k is even, (90) and (91) inthe last line since N ≥ N .Notice that (cid:98) µ (2 Z ) = e − V ( (cid:98) b ( N )) (cid:80) (cid:98) x − i = (cid:98) x e − [ V ( i ) − V ( (cid:98) b ( N ))] , with − (cid:88) i = (cid:98) x e − [ V ( i ) − V ( (cid:98) b ( N ))] ≤ | (cid:98) x | N − ε / ≤ ε − (log N ) N − ε / ≤ ε − since N ≥ N , ω ∈ ∆ (3) N and thanks to (87).Moreover, by (88), (cid:80) (cid:98) x − i = θ ( R ) N e − [ V ( i ) − V ( (cid:98) b ( N ))] ≤ ε − (log N ) N − ε ≤ ε − because N ≥ N .Finally, (cid:80) θ ( R ) N − i =0 e − (cid:2) V ( i ) − V (cid:0) β ( R ) N (cid:1)(cid:3) ≤ ε − since ω ∈ ∆ (6 ,R ) N (see (75)). Moreover, (cid:12)(cid:12) V (cid:0)(cid:98) b ( N ) (cid:1) − V (cid:0) β ( R ) N (cid:1)(cid:12)(cid:12) ≤ a . Hence, (cid:98) µ (2 Z ) ≤ ε − e a e − V ( (cid:98) b ( N )) . Moreover, (cid:98) µ (cid:0)(cid:98) b ( N ) (cid:1) ≥ e − V ( (cid:98) b ( N )) since (cid:98) x < (cid:98) b ( N ) < (cid:98) x , and (cid:98) b ( N ) is even by (77), so by (85), (cid:98) ν (cid:0)(cid:98) b ( N ) (cid:1) = (cid:98) µ (cid:0)(cid:98) b ( N ) (cid:1) / (cid:98) µ (2 Z ) ≥ ε e − a / 3. This, (92) and (93) give for N ≥ N , ∀ ω ∈ ∆ N ∩ ∆ ( R ) N , ∀ n ∈ (cid:2) N − ε / , N (cid:3) ∩ (2 N ) , ∀ j ∈ { , . . . , r } , P y j ω (cid:2) Z n = (cid:98) b ( N ) (cid:3) ≥ ε e − a / . The proof is similar for ω ∈ ∆ N ∩ ∆ ( L ) N by symmetry. This, combined with Lemma5.4, ends (62) with c ( δ ) = ε e − a / > ε ( δ ) = ε / 3. To prove that this remainstrue if (2 Z ) r and 2 N are replaced respectively by (2 Z + 1) r and 2 N + 1, we just condition P y j ω (cid:2) Z n = (cid:98) b ( N ) (cid:3) by Z , and apply the Markov property and (62) to ( y ± , . . . , y r ± (cid:4) Acknowledgement A part of this work was done while AD and NG were visiting Brest.We thank ANR MEMEMO 2 (ANR-10-BLAN-0125) and the LMBA, University of Brestfor its hospitality. We are grateful to an anonymous referee for comments which helpedimprove the presentation of the paper. References [A05] Andreoletti, P. (2005) Alternative proof for the localization of Sinai’s walk. J. Stat. Phys. , 883–933.[AD15] Andreoletti, P. and Devulder, A. (2015) Localization and number of visited valleys fora transient diffusion in random environment. Electron. J. Probab. , no 56, 1–58.[BSP12] Barlow, M., Peres, Y. and Sousi, P. (2012) Collisions of random walks. Ann. Inst. H.Poincar´e Probab. Stat. , no 4, 922–946.[BF08] Bovier, A. and Faggionato, A. (2008) Spectral analysis of Sinai’s walk for small eigen-values. Ann. Probab. , 198–254.[B86] Brox, Th. (1986) A one-dimensional diffusion process in a Wiener medium. Ann. Probab. , 1206–1218.[CC12] Campari, R. and Cassi, D. (2012) Random collisions on branched networks: How simulta-neous diffusion prevents encounters in inhomogeneous structures. Physical Review E .[C05] Cheliotis, D. (2005) Diffusion in random environment and the renewal theorem. Ann.Probab. , 1760–1781.[C08] Cheliotis, D. (2008) Localization of favorite points for diffusion in a random environment. Stoch. Proc. Appl. , 1159–1189.[C01] Chung, K. L. (2001) A course in probability theory. Third edition, Academic Press, Inc.,San Diego.[DR86] Deheuvels, P. and R´ev´esz, P. (1986) Simple random walk on the line in random environ-ment. Probab. Theory Related Fields , 215–230.[DGPS07] Dembo, A., Gantert, N., Peres, Y. and Shi, Z. (2007) Valleys and the maximum localtime for random walk in random environment. Probab. Theory Related Fields , 443–473.[D14] Devulder, A. (2016) Persistence of some additive functionals of Sinai’s walk. Ann. Inst. H.Poincar´e Probab. Stat. , no 3, 1076–1105.[DGP18] Devulder, A., Gantert, N. and P`ene, F. (2018+) Arbitrary many walkers meet infinitelyoften in a subballistic random environment. In preparation.[DS84] Doyle, P. G. and Snell, E. J. (1984) Probability: Random walks and Electrical Networks.Carus Math. Monographs , Math. Assoc. Amer., Washington DC.[DE51] Dvoretzky, A. and Erd¨os, P. (1951) Some problems on random walk in space, Proceedingsof the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, 353–367,University of California Press, Berkeley and Los Angeles.[F15] Freire, M.V. (2015) Application of Moderate Deviation Techniques to Prove Sinai Theoremon RWRE. J. Stat. Phys. (2) 357–370.[Ga13] Gallesco, C. (2013) Meeting time of independent random walks in random environment. ESAIM Probab. Stat. , 257–292. OLLISIONS OF SEVERAL WALKERS IN RECURRENT RANDOM ENVIRONMENTS 35 [GKP14] Gantert, N., Kochler M. and P`ene, F. (2014) On the recurrence of some random walksin random environment. ALEA , 483–502.[G84] Golosov, A. O. (1984) Localization of random walks in one-dimensional random environ-ments. Commun. Math. Phys. , 491–506.[H65] Hirsch, W. M. (1965) A strong law for the maximum cumulative sum of independent randomvariables. Comm. Pure Appl. Math. , 109–127.[HS98] Hu, Y. and Shi, Z. (1998) The limits of Sinai’s simple random walk in random environment. Ann. Probab. , 1477–1521.[HS04] Hu, Y. and Shi, Z. (2004) Moderate deviations for diffusions with Brownian potentials. Ann.Probab. , 3191–3220.[KTT89] Kawazu, K., Tamura, Y. and Tanaka, H. (1989) Limit Theorems for One-DimensionalDiffusions and Random Walks in Random Environments. Probab. Theory Related Fields ,501–541.[KS64] Kochen, S. P. and Stone C. J. (1964) A note on the Borel-Cantelli lemma. Illinois J.Math. , 248–251.[KMT75] Koml´os, J., Major, P. and Tusn´ady, G. (1975) An approximation of partial sums ofindependent RV’s and the sample df. I, Wahrsch verw Gebiete/Probability Theory and RelatedFields , 111–131.[KP04] Krishnapur, M. and Peres, Y. (2004) Recurrent graphs where two independent randomwalks collide infinitely often. Electron. J. Probab. , 72–81.[LL10] Lawler, G. F. and Limic, V. (2010) Random walk: a modern introduction, CambridgeUniversity Press, Cambridge.[N64] Neveu, J. (1964) Bases math´ematiques du calcul des probabilit´es, Masson et Cie, ´Editeurs,Paris.[NP89] Neveu J. and Pitman J. (1989) Renewal property of the extrema and tree property of theexcursion of a one-dimensional Brownian motion. S´eminaire de Probabilit´es XXIII, LectureNotes in Math. , 239–247, Springer, Berlin.[P21] P´olya, G (1921) ¨Uber eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrtim Straßennetz. Math. Ann. , 149–160.[P84] P´olya, G (1984) Collected papers, Vol IV. Edited by Gian-Carlo Rota, M. C. Reynolds andR. M. Shortt. MIT Press , Cambridge, Massachusetts, 1984.[SZ07] Shi, Z. and Zindy, O. (2007) A weakness in strong localization for Sinai’s walk. Ann. Probab. , 1118–1140.[S82] Sinai, Ya. G. (1982) The limiting behavior of a one-dimensional random walk in a randommedium. Th. Probab. Appl. , 256–268.[S75] Solomon, F. (1975) Random walks in a random environment. Ann. Probab. , 1–31.[S76] Spitzer, F. (1976) Principles of random walk, Graduate Texts in Mathematics, Vol. 34,Springer-Verlag, New York-Heidelberg, Second edition.[Z01] Zeitouni, O. (2004) Lecture notes on random walks in random environment. ´Ecole d’´et´e deprobabilit´es de Saint-Flour 2001 . Lecture Notes in Math. , 189–312. Springer, Berlin. Laboratoire de Math´ematiques de Versailles, UVSQ, CNRS, Universit´e Paris-Saclay,78035 Versailles, France. E-mail address : [email protected] Technische Universit¨at M¨unchen, Fakult¨at f¨ur Mathematik, 85748 Garching, Germany E-mail address : [email protected] Universit´e de Brest and Institut Universitaire de France, LMBA, UMR CNRS 6205, 29238Brest cedex, France E-mail address ::