Active random walks in one and two dimensions

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.

Automated summary

This study investigates active lattice walks, biased continuous time random walks that undergo orientational diffusion between lattice directions in one and two spatial dimensions. The authors analyze the occupation probability, large deviation free-energy function, and moments and cumulants of displacements of an active particle in one and two dimensions. Exact results show persistent cross-correlations between motion in the x and y directions in two dimensions, and the large deviation function exhibits two regimes with different diffusive behaviors for active particles. Simulation results confirm the analytical findings.