Discrete space-time resetting model: Application to first-passage and transmission statistics

TL;DR

This paper develops a discrete renewal framework for lattice random walks with resetting, deriving explicit formulas and analyzing first-passage and transmission behaviors in bounded and unbounded domains.

Contribution

It introduces a novel discrete renewal equation approach for resetting random walks and applies it to various boundary conditions and biased dynamics, connecting to continuous diffusion results.

Findings

Resetting probability affects first-passage times non-monotonically.

Transmission probability between walkers varies non-monotonically with resetting.

The formalism recovers known continuous diffusion results in the limit.

Abstract

We consider the dynamics of lattice random walks with resetting. The walker moving randomly on a lattice of arbitrary dimensions resets at every time step to a given site with a constant probability $r$. We construct a discrete renewal equation and present closed-form expressions for different quantities of the resetting dynamics in terms of the underlying reset-free propagator or Green's function. We apply our formalism to the biased random walk dynamics in one-dimensional unbounded space and show how one recovers in the continuous limits results for diffusion with resetting. The resetting dynamics of biased random walker in one-dimensional domain bounded with periodic and reflecting boundaries is also analyzed. Depending on the bias the first-passage probability in periodic domain shows multi-fold non-monotonicity as $r$ is varied. Finally, we apply our formalism to study the transmission dynamics of two lattice walkers with resetting in one-dimensional domain bounded by periodic and reflecting boundaries. The probability of a definite transmission between the walkers shows non-monotonic behavior as the resetting probabilities are varied.