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

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). This is equivalent to the transition between recurrent and transient transport on the tree, with the mean first passage time (MFPT) to be absorbed by an open terminal node switching from a finite value to infinity. 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. In addition, we show that there is a phase transition at a critical velocity $v_c^{**}(z)$ such that $r^{**}>0$ for $v>v_c^{**}$ and $r^{**}=0$ for $v<v_c^{**}$. We then identify a critical velocity $v^*(z)$ for a phase transition in a second optimal resetting rate $r^*$ that minimizes the MFPT 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 $v_c^*(z)<v_c^{**}(z)<v_c(z)$ 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.