The local time of a random walk on growing hypercubes

TL;DR

This paper investigates a reversible random walk in a random environment on growing hypercubes, focusing on local time behavior near specific surfaces and deriving its limit law.

Contribution

It introduces a construction of environment ensuring the walk is confined to certain surfaces, and analyzes the local time distribution and its limit law in this setting.

Findings

Local time near surfaces is governed by the reversible measure.

The walk cannot be trapped at a point but on a surface.

Limit law of local time as a process on surfaces is established.

Abstract

We study a random walk in a random environment (RWRE) on $\Z^d$, $1 \leq d < +\infty$. The main assumptions are that conditionned on the environment the random walk is reversible. Moreover we construct our environment in such a way that the walk can't be trapped on a single point like in some particular RWRE but in some specific d-1 surfaces. These surfaces are basic surfaces with deterministic geometry. We prove that the local time in the neighborhood of these surfaces is driven by a function of the (random) reversible measure. As an application we get the limit law of the local time as a process on these surfaces.