First Order Transition for the Optimal Search Time of Lévy Flights with Resetting

TLDR

The search problem involves an immobile target at the origin and a searcher performing discrete jumps from an arbitrary distribution, with efficiency measured by the mean first passage time to the target. The study analytically investigates an intermittent one‑dimensional search process. The model consists of a searcher making independent jumps from a distribution f(η) and, with probability r, resetting to the starting point x₀. Analytically, the mean first passage time is obtained for any jump distribution, and for Lévy flights it exhibits a global minimum in the (μ,r) plane whose optimal parameters shift discontinuously at a critical starting distance, a first‑order transition confirmed by simulations.

Abstract

We study analytically an intermittent search process in one dimension. There is an immobile target at the origin and a searcher undergoes a discrete time jump process starting at x_{0}≥0, where successive jumps are drawn independently from an arbitrary jump distribution f(η). In addition, with a probability 0≤r<1, the position of the searcher is reset to its initial position x_{0}. The efficiency of the search strategy is characterized by the mean time to find the target, i.e., the mean first passage time (MFPT) to the origin. For arbitrary jump distribution f(η), initial position x_{0} and resetting probability r, we compute analytically the MFPT. For the heavy-tailed Lévy stable jump distribution characterized by the Lévy index 0<μ<2, we show that, for any given x_{0}, the MFPT has a global minimum in the (μ,r) plane at (μ^{*}(x_{0}),r^{*}(x_{0})). We find a remarkable first-order phase transition as x_{0} crosses a critical value x_{0}^{*} at which the optimal parameters change discontinuously. Our analytical results are in good agreement with numerical simulations.