Biased random walk on supercritical percolation: Anomalous fluctuations in the ballistic regime

TL;DR

This paper investigates biased random walks on supercritical percolation clusters, revealing anomalous polynomial fluctuations in the ballistic regime for certain parameters, extending previous understanding of the walk's behavior.

Contribution

It establishes the existence of anomalous polynomial fluctuations with exponent  for  in the ballistic regime, building on prior percolation estimates.

Findings

Fluctuations are of anomalous polynomial order for  in (1,2)

Ballistic regime exhibits non-zero speed with anomalous fluctuations

Central limit theorem holds for  > 2

Abstract

We study biased random walk on the infinite connected component of supercritical percolation on the integer lattice $\mathbb{Z}^d$ for $d\geq 2$. For this model, Fribergh and Hammond showed the existence of an exponent $\gamma$ such that: for $\gamma<1$, the random walk is sub-ballistic (i.e. has zero velocity asymptotically), with polynomial escape rate described by $\gamma$; whereas for $\gamma>1$, the random walk is ballistic, with non-zero speed in the direction of the bias. They moreover established, under the usual diffusive scaling about the mean distance travelled by the random walk in the direction of the bias, a central limit theorem when $\gamma>2$. In this article, we explain how Fribergh and Hammond's percolation estimates further allow it to be established that for $\gamma\in(1,2)$ the fluctuations about the mean are of an anomalous polynomial order, with exponent given by $\gamma^{-1}$.