Optimizing leapover lengths of Lévy flights with resetting

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 r. On the other hand, great relevance is encapsulated by the leapover length l = x_r - 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 ℓ_b(r) if the mean jump length is finite. We compute exactly ℓ_b(r) and determine the condition under which resetting allows for nontrivial optimization, i.e., for the existence of r∗ such that ℓ_b(r∗) is minimal and smaller than the average leapover of the single jump.