The Parabolic Anderson Model on a Galton–Watson Tree Revisited

In den Hollander et al. (The parabolic Anderson model on a Galton-Watson tree, to appear in in and out of equilibrium 3: celebrating Vladas Sidoravicius. Progress in probability, Birkhäuser, Basel, 2021) 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 den Hollander et al. (The parabolic Anderson model on a Galton-Watson tree, to appear in in and out of equilibrium 3: celebrating Vladas Sidoravicius. Progress in probability, Birkhäuser, Basel, 2021) 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.