The Parabolic Anderson Model on a Galton-Watson tree revisited

TL;DR

This paper extends the analysis of the parabolic Anderson model on Galton-Watson trees to unbounded degree distributions, identifying conditions for asymptotic behavior and solution concentration on minimal degree subtrees.

Contribution

It generalizes previous results by removing the bounded degree assumption, providing conditions for large-time asymptotics with unbounded degrees.

Findings

Identified the weakest tail condition for degree distribution

Extended asymptotic analysis to unbounded degree cases

Demonstrated solution concentration on minimal degree subtrees

Abstract

In [1] a detailed analysis was given of the large-time asymptotics of the total mass of the solution to the parabolic Anderson model on a supercritical Galton-Watson random tree with an i.i.d. random potential whose marginal distribution is double-exponential. Under the assumption that the degree distribution has bounded support, two terms in the asymptotic expansion were identified under the quenched law, i.e., conditional on the realisation of the random tree and the random potential. The second term contains a variational formula indicating that the solution concentrates on a subtree with minimal degree according to a computable profile. The present paper extends the analysis to degree distributions with unbounded support. We identify the weakest condition on the tail of the degree distribution under which the arguments in [1] can be pushed through. To do so we need to control the occurrence of large degrees uniformly in large subtrees of the Galton-Watson tree.