On the return probability of the simple random walk on Galton-Watson trees
Probability
2025-01-22 v2 Mathematical Physics
math.MP
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.
Keywords
Cite
@article{arxiv.2402.01600,
title = {On the return probability of the simple random walk on Galton-Watson trees},
author = {Peter Müller and Jakob Stern},
journal= {arXiv preprint arXiv:2402.01600},
year = {2025}
}
Comments
23 pages. To appear in Electron. J. Probab. Changes in v2: Theorem 1.3 strengthened due to an improvement in Lemma 3.6