Maximum and records of random walks with stochastic resetting

TL;DR

This paper investigates the extreme value statistics and record-breaking behavior of symmetric random walks with stochastic resetting, revealing universal scaling laws and exact solutions for specific cases, extending understanding beyond traditional regimes.

Contribution

It introduces a diffusive scaling regime for random walks with resetting, providing new asymptotic laws and exact solutions for certain distributions, advancing the theoretical understanding of extremes and records.

Findings

Scaling laws interpolate between half-Gaussian and Gumbel distributions.

Exact solutions obtained for exponential step length and Polya lattice walks.

Heuristic analysis extends results to other distributions.

Abstract

We revisit the statistics of extremes and records of symmetric random walks with stochastic resetting, extending earlier studies in several directions. We put forward a diffusive scaling regime (symmetric step length distribution with finite variance, weak resetting probability) where the maximum of the walk and the number of its records up to discrete time $n$ become asymptotically proportional to each other for single typical trajectories. Their distributions obey scaling laws ruled by a common two-parameter scaling function, interpolating between a half-Gaussian and a Gumbel law. The exact solution of the problem for the symmetric exponential step length distribution and for the simple Polya lattice walk, as well as a heuristic analysis of other distributions, allow a quantitative study of several facets of the statistics of extremes and records beyond the diffusive scaling regime.