Parabolic Anderson model on critical Galton-Watson trees in a Pareto environment

TL;DR

This paper studies the long-term behavior of the parabolic Anderson model on critical Galton-Watson trees with Pareto-distributed potential, revealing localization at one or two sites and asymptotic properties influenced by the tree's variable degrees.

Contribution

It extends the analysis of the parabolic Anderson model to critical Galton-Watson trees, incorporating variable vertex degrees and identifying localization and asymptotics in this setting.

Findings

Solution concentrates at a single site with high probability as t→∞

Solution localizes at two sites almost surely as t→∞

Asymptotic behavior of localization sites and total mass identified

Abstract

The parabolic Anderson model is the heat equation with some extra spatial randomness. In this paper we consider the parabolic Anderson model with i.i.d. Pareto potential on a critical Galton-Watson tree conditioned to survive. We prove that the solution at time $t$ is concentrated at a single site with high probability and at two sites almost surely as $t \to \infty$. Moreover, we identify asymptotics for the localisation sites and the total mass, and show that the solution $u(t,v)$ at a vertex $v$ can be well-approximated by a certain functional of $v$. The main difference with earlier results on $\mathbb{Z}^d$ is that we have to incorporate the effect of variable vertex degrees within the tree, and make the role of the degrees precise.