Biased random walk on the critical curve of dynamical percolation

TL;DR

This paper investigates the behavior of biased random walks on dynamical percolation in multi-dimensional lattices, providing detailed asymptotic speed expansions and monotonicity results at the critical curve.

Contribution

It offers a second order expansion for the asymptotic speed and proves the speed's monotonic increase on the critical curve for dimensions two and higher.

Findings

Derived a second order asymptotic speed expansion.

Proved monotonic increase of speed on the critical curve for d ≥ 2.

Utilized environment viewed from the walker and coupling techniques.

Abstract

We study biased random walks on dynamical percolation in $\mathbb{Z}^d$, which were recently introduced by Andres et al. We provide a second order expansion for the asymptotic speed and show for $d \ge 2$ that the speed of the biased random walk on the critical curve is eventually monotone increasing. Our methods are based on studying the environment seen from the walker as well as a combination of ergodicity and several couplings arguments.