  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.
