Random walk on a perturbation of the infinitely-fast mixing interchange process

TL;DR

This paper studies a random walk in a dynamic environment modeled by an interchange process, showing that for large enough environment refresh rates, the walk's velocity aligns closely with the environment's average drift, extending previous results to higher dimensions and more general settings.

Contribution

It extends prior work by analyzing random walks in interchange process environments across all dimensions, with arbitrary transition laws, and demonstrates velocity convergence to the annealed drift for large refresh rates.

Findings

The walk's empirical velocity converges to the annealed drift as environment refresh rate increases.

The results hold in any dimension $d\,\geq\,1$ and for arbitrary transition laws.

Velocity remains close to the drift, not just in direction but also in magnitude.

Abstract

We consider a random walk in dimension $d\geq 1$ in a dynamic random environment evolving as an interchange process with rate $\gamma>0$. We only assume that the annealed drift is non-zero. We prove that the empirical velocity of the walker $X_t/t$ eventually lies in an arbitrary small ball around the annealed drift if we choose $\gamma$ large enough. This statement is thus a perturbation of the case $\gamma =+\infty$ where the environment is refreshed between each step of the walker. We extend three-way part of the results of HS15, where the environment was given by the $1$ dimensional exclusion process: $(i)$ We deal with any dimension $d\geq 1$; $(ii)$ Each particle of the interchange process carries a transition vector chosen according to an arbitrary law $\mu$; $(iii)$ We show that $X_t/t$ is not only in the same direction of the annealed drift, but that it is also close to it.