Random walk delayed on percolation clusters

TL;DR

This paper investigates a continuous-time random walk on a lattice influenced by drift and attraction to large percolation clusters, revealing distinct ballistic and subballistic regimes with different escape behaviors.

Contribution

It introduces a model combining drift and attraction to percolation clusters, identifying phase transitions between ballistic and subballistic regimes with explicit speed and escape rate characterizations.

Findings

Identified ballistic and subballistic regimes based on attraction strength.

Derived explicit formulas for the walk's speed in the ballistic regime.

Analyzed diffusive behavior in the zero drift, weak attraction case.

Abstract

We study a continuous time random walk on the $d$-dimensional lattice, subject to a drift and an attraction to large clusters of a subcritical Bernoulli site percolation. We find two distinct regimes: a ballistic one, and a subballistic one taking place when the attraction is strong enough. We identify the speed in the former case, and the algebraic rate of escape in the latter case. Finally, we discuss the diffusive behavior in the case of zero drift and weak attraction.