title: First Order Transition for the Optimal Search Time of Lévy Flights with Resetting.
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.
authors.affiliation: Institute of Physics, UJ, Reymonta 4, 30-059 Krakow, Poland and Department of Automatics and Biomedical Engineering, AGH, Aleja Mickiewicza 30, 30-059 Krakow, Poland.
authors.forename: Lukasz
authors.initials: L
authors.lastname: Kusmierz
authors.affiliation: Université Paris-Sud, CNRS, LPTMS, UMR 8626, Orsay F-91405, France.
authors.forename: Satya N
authors.initials: SN
authors.lastname: Majumdar
authors.affiliation: Raman Research Institute, Bangalore 560080, India.
authors.forename: Sanjib
authors.initials: S
authors.lastname: Sabhapandit
authors.affiliation: Université Paris-Sud, CNRS, LPTMS, UMR 8626, Orsay F-91405, France.
authors.forename: Grégory
authors.initials: G
authors.lastname: Schehr