The annealed parabolic Anderson model on a regular tree

TL;DR

This paper derives the asymptotic behavior of the total mass of the parabolic Anderson model on a regular tree under the annealed law, revealing differences from the quenched case through a novel approach involving local times and large deviations.

Contribution

It introduces a new method to analyze the annealed asymptotics of the parabolic Anderson model on trees, contrasting with previous quenched results.

Findings

Annealed asymptotic expansion differs from quenched expansion.

New approach controls local times via backbone conditioning.

Asymptotic expansion obtained through large deviation principles.

Abstract

We study the total mass of the solution to the parabolic Anderson model on a regular tree with an i.i.d. random potential whose marginal distribution is double-exponential. In earlier work we identified two terms in the asymptotic expansion for large time of the total mass under the quenched law, i.e., conditional on the realisation of the random potential. In the present paper we do the same for the annealed law, i.e., averaged over the random potential. It turns out that the annealed expansion differs from the quenched expansion. The derivation of the annealed expansion is based on a new approach to control the local times of the random walk appearing in the Feynman-Kac formula for the total mass. In particular, we condition on the backbone to infinity of the random walk, truncate and periodise the infinite tree relative to the backbone to obtain a random walk on a finite subtree with a specific boundary condition, employ the large deviation principle for the empirical distribution of Markov renewal processes on finite graphs, and afterwards let the truncation level tend to infinity to obtain an asymptotically sharp asymptotic expansion.