Return probability on Bienaymé-Galton-Watson trees and spectral asymptotics of sparse Erdős-Rényi random graphs

We derive an upper bound for the annealed return probability for the simple random walk on supercritical Bienaymé-Galton-Watson trees. The bound decays subexponentially in time $t$ with $t^{1\mathrm{/}3}$ in the exponent. It is valid for all offspring distributions with a finite first moment and is optimal whenever the offspring distribution does not exclude leaves or linear pieces in the tree. This solves completely the case left open by Piau [Ann. Probab. 26, 1016-1040 (1998)]. In the special case of a Poissonian offspring distribution we apply this upper bound to deduce a Lifshits tail for the empirical eigenvalue distribution of the graph Laplacian on supercritical Erdős-Rényi random graphs with finite mean degree.