The parabolic Anderson model on a Galton-Watson tree with normalised Laplacian

TL;DR

This paper extends the analysis of the parabolic Anderson model on Galton-Watson trees to the degree-normalised Laplacian, revealing that while the leading asymptotics remain unchanged, the second-order corrections differ due to spectral property changes.

Contribution

It introduces the study of the parabolic Anderson model with a normalized Laplacian on Galton-Watson trees, highlighting differences in spectral properties and second-order asymptotics.

Findings

Leading order asymptotics remain the same with normalized Laplacian.

Second-order correction differs from the unnormalized case.

Optimizer of the variational formula is an infinite tree with minimal degrees.

Abstract

In earlier work by den Hollander, K\"onig, and dos Santos, the asymptotics of the total mass of the solution to the parabolic Anderson model was studied on an almost surely infinite Galton-Watson tree with an i.i.d. potential having a double-exponential distribution. The second-order contribution to this asymptotics was identified in terms of a variational formula that gives information about the local structure of the region where the solution is concentrated. The present paper extends this work to the degree-normalised Laplacian. The normalisation causes the Laplacian to be non-symmetric and which leads to different spectral properties. We find that the leading order asymptotics of the total mass remains the same, while the second-order correction coming from the variational formula is different. We also find that the optimiser of the variational formula is again an infinite tree with minimal degrees. Both of these results are shown to hold under much milder conditions than for the regular Laplacian.