Drift-diffusion on a Cayley tree with stochastic resetting: the localization–delocalization transition

Abstract

In this paper we develop the theory of drift-diffusion on a semi-infinite Cayley tree with stochastic resetting. In the case of a homogeneous tree with a closed terminal node and no resetting, it is known that the system undergoes a classical localization–delocalization (LD) transition at a critical mean velocity v c = −(D/L)ln(z − 1) where D is the diffusivity, L is the branch length and z is the coordination number of the tree. If v < v c then the steady state concentration at the terminal node is non-zero (drift-dominated localized state), whereas it is zero when v > v c (diffusion-dominated delocalized state). Here we show how the LD transition provides a basic framework for understanding analogous phase transitions in optimal resetting rates. First, we establish the existence of an optimal resetting rate r ∗∗(z) that maximizes the steady-state solution at a closed terminal node with respect to r. In addition, we show that there is a phase transition at a critical velocity such that r ∗∗ > 0 for and r ∗∗ = 0 for . We then identify a critical velocity for a phase transition in a second optimal resetting rate r* that minimizes the mean first passage time to be absorbed by an open terminal node. Previous results for the semi-infinite line are recovered on setting z = 2. The critical velocity of the LD transition provides an upper bound for the other critical velocities such that for all finite z. Only v c(z) has a simple universal dependence on the coordination number z. We end by considering the combined effects of quenched disorder and stochastic resetting.