Returns to the origin of the P\'olya walk with stochastic resetting

TL;DR

This paper analyzes the stochastic properties of a one-dimensional Pólya walk with resetting, revealing linear growth of return and reset counts, a large deviation structure, and a renewal process description of return intervals.

Contribution

It provides explicit calculations of joint cumulants, large deviation functions, and uncovers a renewal process structure for return times in a reset random walk.

Findings

Joint cumulants grow linearly with time.

Fluctuations are described by a smooth large deviation function.

Return intervals form a renewal process with a dressed distribution.

Abstract

We consider the simple random walk (or P\'olya walk) on the one-dimensional lattice subject to stochastic resetting to the origin with probability $r$ at each time step. The focus is on the joint statistics of the numbers ${\mathcal{N}}_t^{\times}$ of spontaneous returns of the walker to the origin and ${\mathcal{N}}_t^{\bullet}$ of resetting events up to some observation time $t$. These numbers are extensive in time in a strong sense: all their joint cumulants grow linearly in $t$, with explicitly computable amplitudes, and their fluctuations are described by a smooth bivariate large deviation function. A non-trivial crossover phenomenon takes place in the regime of weak resetting and late times. Remarkably, the time intervals between spontaneous returns to the origin of the reset random walk form a renewal process described in terms of a single `dressed' probability distribution. These time intervals are probabilistic copies of the first one, the `dressed' first-passage time. The present work follows a broader study, covered in a companion paper, on general nested renewal processes.