The infinite valley for a recurrent random walk in random environment

TL;DR

This paper studies a one-dimensional recurrent random walk in a random environment, demonstrating convergence of empirical distributions to a limit law related to an infinite valley, and analyzing local times and self-intersections.

Contribution

It establishes weak convergence of empirical distributions to a limit law associated with the infinite valley, extending Golosov's construction, and determines constants for local time behavior.

Findings

Weak convergence of empirical distributions to the stationary law.

Convergence results for maximal local time and self-intersection local time.

Exact constants for the upper limits of local times.

Abstract

We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the - suitably centered - empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the RWRE and also determine the exact constant in the almost sure upper limit of the maximal local time.