Fluctuations of the quenched mean of a planar random walk in an i.i.d. random environment with forbidden direction

TL;DR

This paper establishes a functional central limit theorem for the quenched expected position of a two-dimensional random walk in an i.i.d. environment with a forbidden backward move, focusing on fluctuations of the quenched mean.

Contribution

It introduces a novel application of a Martingale CLT to analyze quenched mean fluctuations in a constrained planar random walk environment.

Findings

Proves a functional CLT for the quenched expected position

Identifies conditions under which the CLT applies in this setting

Provides a methodological framework for similar stochastic processes

Abstract

We consider an i.i.d. random environment with a strong form of transience on the two dimensional integer lattice. Namely, the walk always moves forward in the y-direction. We prove a functional CLT for the quenched expected position of the random walk indexed by its level crossing times. We begin with a variation of the Martingale Central Limit Theorem. The main part of the paper checks the conditions of the theorem for our problem.