Optimizing Leapover Lengths of L\'evy Flights with Resetting

Abstract

We consider a one-dimensional search process under stochastic resetting conditions. A target is located at $b\geq0$ 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 $\tau$. On the other hand, great relevance is encapsulated by the leapover length $l = x_{\tau} - b$, 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 $\ell_b(r)$ if the mean jump length is finite. We compute exactly $\ell_b(r)$ and determine the condition under which resetting allows for nontrivial optimization, i.e., for the existence of $r^*$ such that $\ell_b(r^*)$ is minimal and smaller than the average leapover of the single jump.