Two-sided estimates for two-point local time tails of branching random walks: a Green function criterion

Abstract

This paper is devoted to the study of the joint tails of local times at two distinct sites for branching random walks (BRWs). Our principle results establish a general Green-function criterion for two-sided exponential tail bounds with a common spatial factor given by a three-point Green function convolution, and further identify a strict inequality that must be satisfied by the corresponding exponential constants. In particular, we verify this criterion for subcritical BRWs on finitely generated groups, for critical BRWs on non-elementary hyperbolic groups, and for critical BRWs on $\mathbb{Z}^d$ with $d\geq 5$. As applications, we revisit the growth rate of visited sites on spheres in hyperbolic groups obtained by Sidoravicius et al. [Commun. Pure Appl. Math. 2023], and also establish lower bounds for the probabilities of visiting finite sets in Euclidean lattices.