Optimizing leapover lengths of Lévy flights with resetting

Abstract

We consider a one-dimensional search process under stochastic resetting conditions. A target is located at b≥0 and a searcher, starting from the origin, performs a discrete-time random walk with independent jumps drawn from a heavy-tailed distribution. Before each jump, there is a given probability r of restarting the walk from the initial position. The efficiency of a “myopic search”—in which the search stops upon crossing the target for the first time—is usually characterized in terms of the first-passage time τ. On the other hand, great relevance is encapsulated by the leapover length l=xτ−b, which measures how far from the target the search ends. For symmetric heavy-tailed jump distributions, in the absence of resetting the average leapover is always infinite. Here we show instead that resetting induces a finite average leapover ℓb(r) if the mean jump length is finite. We compute exactly ℓb(r) and determine the condition under which resetting allows for nontrivial optimization, i.e., for the existence of r* such that ℓb(r*) is minimal and smaller than the average leapover of the single jump. Published by the American Physical Society 2024