Biased random walk on dynamical percolation

TL;DR

This paper analyzes biased random walks on dynamical percolation in multiple dimensions, establishing fundamental probabilistic laws, the Einstein relation, and revealing complex speed behaviors depending on the bias and dimension.

Contribution

It introduces a detailed study of biased random walks on dynamical percolation, proving laws of large numbers, invariance principles, and identifying regimes of speed behavior separated by a critical curve.

Findings

Law of large numbers and invariance principle established

Einstein relation verified for the model

Speed behavior varies with bias and dimension, with distinct regimes

Abstract

We study biased random walks on dynamical percolation on $\mathbb{Z}^d$. We establish a law of large numbers and an invariance principle for the random walk using regeneration times. Moreover, we verify that the Einstein relation holds, and we investigate the speed of the walk as a function of the bias. While for $d=1$ the speed is increasing, we show that in general this fails in dimension $d \geq 2$. As our main result, we establish two regimes of parameters, separated by an explicit critical curve, such that the speed is either eventually strictly increasing or eventually strictly decreasing. This is in sharp contrast to the biased random walk on a static supercritical percolation cluster, where the speed is known to be eventually zero.