First passage under restart for discrete space and time: application to one dimensional confined lattice random walks

TL;DR

This paper develops a framework for analyzing discrete space and time first passage processes under restart, providing theoretical tools and criteria for when restart is beneficial, with applications to one-dimensional lattice random walks.

Contribution

It introduces a general method for computing moments and probability densities of discrete first passage times under restart, filling a gap in existing continuous-time studies.

Findings

Restart can significantly reduce mean first passage times.

Theoretical results match numerical simulations closely.

Criteria for beneficial restart are established.

Abstract

First passage under restart has recently emerged as a conceptual framework to study various stochastic processes under restart mechanism. Emanating from the canonical diffusion problem by Evans and Majumdar, restart has been shown to outperform the completion of many first passage processes which otherwise would take longer time to finish. However, most of the studies so far assumed continuous time underlying first passage time processes and moreover considered continuous time resetting restricting out restart processes broken up into synchronized time steps. To bridge this gap, in this paper, we study discrete space and time first passage processes under discrete time resetting in a general set-up. We sketch out the steps to compute the moments and the probability density function which is often intractable in the continuous time restarted process. A criterion that dictates when restart remains beneficial is then derived. We apply our results to a symmetric and a biased random walker (RW) in one dimensional lattice confined within two absorbing boundaries. Numerical simulations are found to be in excellent agreement with the theoretical results. Our method can be useful to understand the effect of restart on the spatiotemporal dynamics of confined lattice random walks in arbitrary dimensions.