First order transition for the optimal search time of L\'evy flights with resetting

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 $x_0\ge 0$, where successive jumps are drawn independently from an arbitrary jump distribution $f(\eta )$. In addition, with a probability $0\le r\le 1$ 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 $f(\eta )$, 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 $({\mu }^{\ast }(x_0),r^{\ast }(x_0))$. We find a remarkable first-order phase transition as $x_0$ crosses a critical value $x_0^{\ast }$ at which the optimal parameters change discontinuously. Our analytical results are in good agreement with numerical simulations.