## Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

### Maximal displacement for bridges of random walks in a random environment

#### Abstract

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{−CnC'n/(ln n)2+o(n/(ln n)2)} with explicit constants C, C'>0.

#### Résumé

Il est bien connu que la distribution d’une marche aléatoire simple sur ℤ, conditionée 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, conditionellement à {S2n=0}, le déplacement maximum maxk≤2n|Sk|, divisé par √n, converge en distribution.

Nous considérons le mème 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’environement a des drifts locaux positifs et negatifs alors cet ordre de grandeur est nκ/(κ+1), où κ>0 dépend de la loi de l’environement. Mais, si l’environement 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{−CnC'n/(ln n)2+o(n/(ln n)2)}.

#### Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 3 (2011), 663-678.

Dates
First available in Project Euclid: 23 June 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1308834854

Digital Object Identifier
doi:10.1214/10-AIHP378

Mathematical Reviews number (MathSciNet)
MR2841070

Zentralblatt MATH identifier
1262.60096

Subjects
Primary: 60K37: Processes in random environments

#### Citation

Gantert, Nina; Peterson, Jonathon. Maximal displacement for bridges of random walks in a random environment. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 3, 663--678. doi:10.1214/10-AIHP378. https://projecteuclid.org/euclid.aihp/1308834854

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