First order transition for the optimal search time of Lévy flights with resetting

Lukasz Kusmierz 1, 2 Satya N. Majumdar 3 Sanjib Sabhapandit 4 Gregory Schehr 3

Physical Review Letters, American Physical Society, 2014, 113, pp.220602

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 $x_0\geq0$, where successive jumps are drawn independently from an arbitrary jump distribution $f(\eta)$. In addition, with a probability $0\leq r \leq1$ 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 $f(\eta)$, initial position $x_0$ and resetting probability $r$, we compute analytically the MFPT. For the heavy-tailed L\'evy stable jump distribution characterized by the L\'evy index $0<\mu < 2$, we show that, for any given $x_0$, the MFPT has a global minimum in the $(\mu,r)$ plane at $(\mu^*(x_0),r^*(x_0))$. 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.

  • 1. Institute of Physics of Krakow
  • 2. Department of Automatics and Biomedical Engineering
  • 3. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
  • 4. Raman Research Institute