Active Random Walks in One and Two Dimensions

TL;DR

This paper provides exact analytical results for active lattice walks in one and two dimensions, revealing persistent cross-correlations and distinct diffusive regimes, supported by kinetic Monte Carlo simulations.

Contribution

It derives exact occupation probabilities, large deviation functions, and moments for active lattice walks, highlighting cross-dimensional correlations and regime transitions.

Findings

Persistent cross-correlations between x and y motions in 2D.

Two distinct diffusive regimes in the large deviation function.

Analytic results validated by kinetic Monte Carlo simulations.

Abstract

We investigate active lattice walks: biased continuous time random walks which perform orientational diffusion between lattice directions in one and two spatial dimensions. We study the occupation probability of an arbitrary site on the lattice in one and two dimensions, and derive exact results in the continuum limit. Next, we compute the large deviation free energy function in both one and two dimensions, which we use to compute the moments and the cumulants of the displacements exactly at late times. Our exact results demonstrate that the cross-correlations between the motion in the $x$ and $y$ directions in two dimensions persist in the large deviation function. We also demonstrate that the large deviation function of an active particle with diffusion displays two regimes, with differing diffusive behaviors. We verify our analytic results with kinetic Monte Carlo simulations of an active lattice walker in one and two dimensions.