This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties.Dieses Dokument darf im Rahmen von § 53 UrhG zum eigenen Gebrauch kostenfrei heruntergeladen, gelesen, gespeichert und ausgedruckt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden.den Hollander, FrankKönig, WolfgangSoares dos Santos, Renato2022-06-302022-06-302020https://oa.tib.eu/renate/handle/123456789/9325https://doi.org/10.34657/8363We 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.eng510Parabolic Anderson modelrandom graphstree-like graphsGalton--Watson treerandom walk in random potentiallarge-time asymptoticsalmost-sure asymptoticseigenvalues of random operatorsReport36 S.