WIAS Preprint No. 2675, (2020)

The parabolic Anderson model on a Galton--Watson tree



Authors

  • den Hollander, Frank
  • König, Wolfgang
    ORCID: 0000-0002-7673-4364
  • Soares dos Santos, Renato

2010 Mathematics Subject Classification

  • 05C80 60H25 82B44

Keywords

  • Parabolic Anderson model, random graphs, tree-like graphs, Galton--Watson tree, random walk in random potential, large-time asymptotics, almost-sure asymptotics, eigenvalues of random operators

DOI

10.20347/WIAS.PREPRINT.2675

Abstract

We study the long-time asymptotics of the total mass of the solution to the parabolic Anderson model ( PAM) on a supercritical Galton-Watson random tree with bounded degrees. We identify the second-order contribution to this asymptotics in terms of a variational formula that gives information about the local structure of the region where the solution is concentrated. The analysis behind this formula suggests that, under mild conditions on the model parameters, concentration takes place on a tree with minimal degree. Our approach can be applied to finite locally tree-like random graphs, in a coupled limit where both time and graph size tend to infinity. As an example, we consider the configuration model or, more precisely, the uniform simple random graph with a prescribed degree sequence.

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