Maximal displacement for bridges of random walks in a random environment

Gantert, Nina; Peterson, Jonathon

Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 3, pp. 663-678.

Il est bien connu que la distribution d'une marche aléatoire simple sur ℤ, conditionnée à retourner à l'origine au temps 2n est indépendante de p=P(S1=1), la probabilité d'un pas vers la droite. De plus, conditionnellement à {S2n=0}, le déplacement maximum maxk≤2n|Sk|, divisé par √n, converge en distribution. Nous considérons le même problème pour les marches transientes en environnement aléatoire sur ℤ. Nous montrons que sous la loi “quenched,” le déplacement maximum pour la marche conditionnée à retourner à l'origine au temps 2n n'est pas toujours de l'ordre de √n. Si l'environnement a des drifts locaux positifs et négatifs alors cet ordre de grandeur est nκ/(κ+1), où κ>0 dépend de la loi de l'environnement. Mais, si l'environnement n'a que des drifts locaux positifs ou nuls, alors cet ordre de grandeur est proche de n. Les preuves fournissent de plus l'ordre de grandeur de Pω(X2n=0). Dans le cas où les drifts locaux sont tous positifs nous montrons que Pω(X2n=0)=exp{-Cn-C'n/(ln n)2+o(n/(ln n)2)}.

It is well known that the distribution of simple random walks on ℤ conditioned on returning to the origin after 2n steps does not depend on p=P(S1=1), the probability of moving to the right. Moreover, conditioned on {S2n=0} the maximal displacement maxk≤2n|Sk| converges in distribution when scaled by √n (diffusive scaling). We consider the analogous problem for transient random walks in random environments on ℤ. We show that under the quenched law Pω (conditioned on the environment ω), the maximal displacement of the random walk when conditioned to return to the origin at time 2n is no longer necessarily of the order √n. If the environment is nestling (both positive and negative local drifts exist) then the maximal displacement conditioned on returning to the origin at time 2n is of order nκ/(κ+1), where the constant κ>0 depends on the law on environments. On the other hand, if the environment is marginally nestling or non-nestling (only non-negative local drifts) then the maximal displacement conditioned on returning to the origin at time 2n is at least n1-ε and at most n/(ln n)2-ε for any ε>0. As a consequence of our proofs, we obtain precise rates of decay for Pω(X2n=0). In particular, for certain non-nestling environments we show that Pω(X2n=0)=exp{-Cn-C'n/(ln n)2+o(n/(ln n)2)} with explicit constants C, C'>0.

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