First passage statistics of Poisson random walks on lattices

TL;DR

This paper derives exact first passage statistics for Poisson random walks on 1D and 2D lattices, providing analytical expressions and validating them with simulations.

Contribution

It presents the first exact expressions for first return probabilities and passage distributions for Poisson random walks on lattices, advancing understanding of their stochastic behavior.

Findings

Exact first return probability in 1D for symmetric and biased walkers.

Laplace transform of occupation probability linked to lattice Green's function.

Analytical results validated with kinetic Monte Carlo simulations.

Abstract

The first passage statistics of a continuous time random walker with Poisson distributed jumps on one and two dimensional infinite lattices is investigated. An exact expression for the probability of first return to the origin in one dimension is derived for a symmetric random walker as well as a biased random walker. The Laplace transform of the occupation probability of a site for a symmetric random walker on a two dimensional lattice is identified with the lattice Green's function for a square lattice. This allows computation of the exact first passage distribution to any arbitrary site on the square lattice in Laplace space. All analytical results are compared with kinetic Monte Carlo simulations of a lattice walker in one and two dimensions.