Maximal displacement for bridges of random walks in a random environment

Gantert, Nina, and Peterson, Jonathon. "Maximal displacement for bridges of random walks in a random environment." Annales de l'I.H.P. Probabilités et statistiques 47.3 (2011): 663-678. <http://eudml.org/doc/240383>.

@article{Gantert2011,
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&gt;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&gt;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'&gt;0.},
author = {Gantert, Nina, Peterson, Jonathon},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random walk in random environment; moderate deviations},
language = {eng},
number = {3},
pages = {663-678},
publisher = {Gauthier-Villars},
title = {Maximal displacement for bridges of random walks in a random environment},
url = {http://eudml.org/doc/240383},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Gantert, Nina
AU - Peterson, Jonathon
TI - Maximal displacement for bridges of random walks in a random environment
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2011
PB - Gauthier-Villars
VL - 47
IS - 3
SP - 663
EP - 678
AB - 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&gt;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&gt;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'&gt;0.
LA - eng
KW - random walk in random environment; moderate deviations
UR - http://eudml.org/doc/240383
ER -