Maximum and records of random walks with stochastic resetting

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.

36 pages, 6 figures

References in corpus (8)

• Universal Record Statistics of Random Walks and Lévy Flights
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Cited by in corpus (11)

• Universal framework for record ages under restart
• Random walks on complex networks under time-dependent stochastic resetting
• Extremal statistics for a resetting Brownian motion before its first-passage time
• Entropy rate of random walks on complex networks under stochastic resetting
• Extreme value statistics of positive recurrent centrally biased random walks
• Stochastic resetting of a population of random walks with resetting-rate-dependent diffusivity
• Survival probability of random walks and Lévy flights with stochastic resetting
• Non-homogeneous random walks with stochastic resetting: an application to the Gillis model
• Replicating a renewal process at random times
• Returns to the origin of the Pólya walk with stochastic resetting
• Full Record Statistics of 1d Random Walks