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arXiv · 2203.01102

Maximum and records of random walks with stochastic resetting

Claude Godrèche·Jean-Marc Luck
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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.

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Claude Godrèche, Jean-Marc Luck. 2022-03-02. Maximum and records of random walks with stochastic resetting. https://doi.org/10.1088/1742-5468%2Fac6d60

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