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 of the simple random walk on supercritical Bienaymé-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 cases left open by Piau [Ann. Probab. 26, 1016-1040 (1998)]. A new feature of our proof is a far-reaching flexibility in the location of regions with bad isoperimetric properties in the tree. It allows to efficiently treat general offspring distributions and is gained from the joint consideration of the random tree and the random walk as it is inherent under the annealed measure. In the special case of a Poissonian offspring distribution we apply the upper bound for the annealed return probability to deduce a Lifshits tail for the empirical eigenvalue distribution of the graph Laplacian on supercritical Erdős--Rényi random graphs with finite mean degree.