On the return probability of the simple random walk on Galton–Watson trees

Abstract

We consider the simple random walk on Galton–Watson trees with supercritical offspring distribution, conditioned on non-extinction. In case the offspring distribution has finite support, we prove an upper bound for the annealed return probability to the root which decays subexponentially in time with exponent 1/3. This exponent is optimal. Our result improves the previously known subexponential upper bound with exponent 1/5 by Piau [Ann. Probab. 26, 1016–1040 (1998)]. For offspring distributions with unbounded support but sufficiently fast decay, our method also yields improved subexponential upper bounds.