Return probability on Bienaym\'e-Galton-Watson trees and spectral asymptotics of sparse Erd\H{o}s-R\'enyi random graphs

We derive an upper bound for the annealed return probability for the simple random walk on supercritical Bienaym\'e-Galton-Watson trees. The bound decays subexponentially in time $t$ with $t^{1/3}$ 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 Erd\H{o}s-R\'enyi random graphs with finite mean degree.