Stochastic Search with Poisson and Deterministic Resetting

TL;DR

This paper studies how resetting strategies, both stochastic and deterministic, affect the efficiency of search processes in different dimensions, revealing optimal reset rates and surprising invariances.

Contribution

It introduces a comprehensive analysis of resetting in multi-dimensional stochastic searches, highlighting new phenomena and optimal strategies for different numbers of searchers.

Findings

Optimal reset rate minimizes search time for small N.

Deterministic resetting can lead to lower search costs than stochastic resetting.

Search time becomes independent of reset time T as 1/T approaches zero.

Abstract

We investigate a stochastic search process in one, two, and three dimensions in which $N$ diffusing searchers that all start at $x_0$ seek a target at the origin. Each of the searchers is also reset to its starting point, either with rate $r$, or deterministically, with a reset time $T$. In one dimension and for a small number of searchers, the search time and the search cost are minimized at a non-zero optimal reset rate (or time), while for sufficiently large $N$, resetting always hinders the search. In general, a single searcher leads to the minimum search cost in one, two, and three dimensions. When the resetting is deterministic, several unexpected feature arise for $N$ searchers, including the search time being independent of $T$ for $1/T\to 0$ and the search cost being independent of $N$ over a suitable range of $N$. Moreover, deterministic resetting typically leads to a lower search cost than in stochastic resetting.