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. case homogeneous closed terminal node and no resetting, it is known that system undergoes classical localization-delocalization (LD) transition at critical mean velocity $v_c= -(D/L)\ln (z-1)$ where $D$ diffusivity, $L$ branch length $z$ coordination number tree. If $v v_c $ (diffusion-dominated delocalized state). This equivalent to between recurrent transient transport tree, first passage time (MFPT) be absorbed by an open switching from finite value infinity. Here show how LD provides basic framework for understanding analogous phase transitions in optimal resetting rates. First, establish existence rate $r^{**}(z)$ maximizes steady-state solution node. addition, there $v_c^{**}(z)$ such $r^{**}>0$ $v>v_c^{**}$ $r^{**}=0$ $v<v_c^{**}$. We identify $v^*(z)$ second $r^*$ minimizes MFPT Previous results line are recovered setting $z=2$. The upper bound other velocities $v_c^*(z)<v_c^{**}(z)<v_c(z)$ all $z$. Only $v_c(z)$ has simple universal dependence end considering combined effects quenched disorder