The speed of a biased random walk on a percolation cluster at high density

TL;DR

This paper investigates how the speed of a biased random walk on a percolation cluster in high-density lattices depends on the percolation parameter, providing a first-order expansion near full density.

Contribution

It introduces a first-order expansion of the walk's speed at high density, revealing how percolation affects the walk's velocity in the lattice.

Findings

Percolation slows down the random walk near full density.

The speed expansion is valid for drifts along lattice components.

The study focuses on the case where the percolation parameter p approaches 1.

Abstract

We study the speed of a biased random walk on a percolation cluster on $\Z^d$ in function of the percolation parameter $p$. We obtain a first order expansion of the speed at $p=1$ which proves that percolating slows down the random walk at least in the case where the drift is along a component of the lattice.