Collisions of Random Walks in Dynamic Random Environments

TL;DR

This paper proves that two independent random walks in a dynamic, stationary environment on a 2D lattice collide infinitely often almost surely, under certain moment conditions, with applications to dynamical percolation.

Contribution

It establishes infinite collisions of independent random walks in dynamic environments, extending previous static results to evolving random conductance models.

Findings

Random walks collide infinitely often in dynamic environments.

Results apply to random walks on dynamical percolation.

Proves collision behavior under second moment assumptions.

Abstract

We study dynamic random conductance models on $\mathbb{Z}^2$ in which the environment evolves as a reversible Markov process that is stationary under space-time shifts. We prove under a second moment assumption that two conditionally independent random walks in the same environment collide infinitely often almost surely. These results apply in particular to random walks on dynamical percolation.