A Sharp Phase Transition for Killed Branching Random Walks with Heavy-Tailed Displacements

Abstract

We study the survival of a branching random walk in a supercritical Galton-Watson tree subject to a deterministic, superlinear killing barrier with heavy-tailed displacements. For a fixed $m>0$, we consider a family of such barriers and kill particles whose ancestral paths fall below the barrier. Under some standing assumptions, we establish a sharp phase transition at $m=4$: the process becomes extinct almost surely for $m<4$, while survives with positive probability for $m>4$. Notably, a natural first-moment heuristic suggests the threshold $m=2$; the true critical value $4$ arises from a deterministic constraint along infinite surviving rays. At the critical value $m=4$, we provide examples showing that the same standing assumptions do not determine the survival behavior. The proofs are based on analysis of infinite surviving rays and embedded trees.