Return probability on Bienaymé-Galton-Watson trees and spectral asymptotics of sparse Erdős-Rényi random graphs
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| Format: | Preprint |
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2026
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| _version_ | 1866912938441310208 |
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| author | Heydenreich, Markus Müller, Peter Terveer, Sara |
| author_facet | Heydenreich, Markus Müller, Peter Terveer, Sara |
| contents | 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 $t^{1/3}$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_00344 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Return probability on Bienaymé-Galton-Watson trees and spectral asymptotics of sparse Erdős-Rényi random graphs Heydenreich, Markus Müller, Peter Terveer, Sara Probability Mathematical Physics 05C81, 60K37, 60J80 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 $t^{1/3}$ 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. |
| title | Return probability on Bienaymé-Galton-Watson trees and spectral asymptotics of sparse Erdős-Rényi random graphs |
| topic | Probability Mathematical Physics 05C81, 60K37, 60J80 |
| url | https://arxiv.org/abs/2603.00344 |