Valleys and the maximum local time for random walk in random environment
| dc.creator | Dembo, Amir | |
| dc.creator | Gantert, Nina | |
| dc.creator | Peres, Yuval | |
| dc.creator | Shi, Zhan | |
| dc.date | 2005-08-29 | |
| dc.date | 2006-03-09 | |
| dc.date.accessioned | 2026-07-07T06:42:52Z | |
| dc.date.available | 2026-07-07T06:42:52Z | |
| dc.description | Let $ξ(n, x)$ be the local time at $x$ for a recurrent one-dimensional random walk in random environment after $n$ steps, and consider the maximum $ξ^*(n) = \max_x ξ(n,x)$. It is known that $\limsup ξ^*(n)/n$ is a positive constant a.s. We prove that $\liminf_n (\log\log\log n)ξ^*(n)/n$ is a positive constant a.s.; this answers a question of P. Révész (1990). The proof is based on an analysis of the {\em valleys /} in the environment, defined as the potential wells of record depth. In particular, we show that almost surely, at any time $n$ large enough, the random walker has spent almost all of its lifetime in the two deepest valleys of the environment it has encountered. We also prove a uniform exponential tail bound for the ratio of the expected total occupation time of a valley and the expected local time at its bottom. | |
| dc.description | 30 pages | |
| dc.identifier | https://arxiv.org/abs/math/0508579 | |
| dc.identifier | http://arxiv.org/abs/math/0508579 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/102200 | |
| dc.subject | Probability | |
| dc.subject | 60K37; 60G50;60J55;60F10 | |
| dc.title | Valleys and the maximum local time for random walk in random environment | |
| dc.type | text |