Returns to the origin of the P\'olya walk with stochastic resetting
Claude Godr\`eche, Jean-Marc Luck

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 at each time step. The focus is on the joint statistics of the numbers of spontaneous returns of the walker to the origin and of resetting events up to some observation time . These numbers are extensive in time in a strong sense: all their joint cumulants grow linearly in , 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.…
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Taxonomy
TopicsDiffusion and Search Dynamics
