Diffusion with two resetting points

Abstract: We study the problem of a target search by a Brownian particle subject to stochastic resetting to a pair of sites. The mean search time is minimized by an optimal resetting rate which does not vary smoothly, in contrast with the well-known single site case, but exhibits a discontinuous transition as the position of one resetting site is varied while keeping the initial position of the particle fixed, or vice-versa. The discontinuity vanishes at a "liquid-gas" critical point in position space. This critical point exists provided that the relative weight $m$ of the further site is comprised in the interval $[2.9028...,8.5603...]$. When the initial position follows the resetting point distribution, a discontinuous transition also exists for the optimal rate as the distance between the resetting points is varied, provided that $m$ exceeds the critical value $m_c=6.6008...$ This setup can be mapped onto an intermittent search problem with switching diffusion coefficients and represents a minimal model for the study of distributed resetting.