Fluctuation bounds for symmetric random walks on dynamic environments via Russo-Seymour-Welsh

TL;DR

This paper establishes lower bounds on the fluctuations of symmetric random walks in dynamic environments using percolation-inspired techniques, specifically the Russo-Seymour-Welsh inequality, in a weak influence regime.

Contribution

It introduces a novel approach to bound fluctuations in dynamic environments by adapting RSW techniques, applicable to Gaussian and Confetti percolation models.

Findings

Lower bounds for fluctuations in 1+1 dimensional dynamic environments.

Applicability to Gaussian fields and Confetti percolation models.

Use of RSW inequality in the context of random walks on dynamic environments.

Abstract

In this article, we prove a lower bound for the fluctuations of symmetric random walks on dynamic random environments in dimension $1 + 1$ in the perturbative regime where the walker is weakly influenced by the environment. We suppose that the random environment is invariant with respect to translations and reflections, satisfies the FKG inequality and a mild mixing condition. The techniques employed are inspired by percolation theory, including a Russo-Seymour-Welsh (RSW) inequality. To exemplify the generality of our results, we provide two families of fields that satisfy our hypotheses: a class of Gaussian fields and Confetti percolation models.