The Brownian Vacancy Driven Walk

TL;DR

This paper studies a lattice walk driven by a Brownian vacancy in a particle-filled lattice, revealing non-Gaussian displacement distributions and recurrence only in two dimensions.

Contribution

It introduces a novel model of vacancy-driven lattice walk and derives exact probability distributions for displacement and steps, highlighting unique non-Gaussian behavior.

Findings

Displacement distributions are non-Gaussian.

Walk is recurrent only in two dimensions.

Derived exact probability distributions for the walk.

Abstract

We investigate the lattice walk performed by a tagged member of an infinite `sea' of particles filling a d-dimensional lattice, in the presence of a Brownian vacancy. Particle-particle exchange is forbidden; the only interaction between them being hard core exclusion. The tagged particle, differing from the others only by its tag, moves only when it exchanges places with the hole. In this sense, it is a lattice walk ``driven'' by the Brownian vacancy. The probability distributions for its displacement and for the number of steps taken, after $n$-steps of the vacancy, are derived. Surprisingly, none of them is a Gaussian! It is shown that the only nontrivial dimension where the walk is recurrent is d=2.