Recurrence and transience of the critical random walk snake in random conductances
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916546351202304 |
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| author | Legrand, Alexandre Sabot, Christophe Schapira, Bruno |
| author_facet | Legrand, Alexandre Sabot, Christophe Schapira, Bruno |
| contents | In this paper we study the recurrence and transience of the $\mathbb{Z}^d$-valued branching random walk in random environment indexed by a critical Bienaymé-Galton-Watson tree, conditioned to survive. The environment is made either of random conductances or of random traps on each vertex. We show that when the offspring distribution is non degenerate with a finite third moment and the environment satisfies some suitable technical assumptions, then the process is recurrent up to dimension four, and transient otherwise. The proof is based on a truncated second moment method, which only requires to have good estimates on the quenched Green's function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_17622 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Recurrence and transience of the critical random walk snake in random conductances Legrand, Alexandre Sabot, Christophe Schapira, Bruno Probability 60K37, 60J80 (Primary) 82B41, 82D30 (Secondary) In this paper we study the recurrence and transience of the $\mathbb{Z}^d$-valued branching random walk in random environment indexed by a critical Bienaymé-Galton-Watson tree, conditioned to survive. The environment is made either of random conductances or of random traps on each vertex. We show that when the offspring distribution is non degenerate with a finite third moment and the environment satisfies some suitable technical assumptions, then the process is recurrent up to dimension four, and transient otherwise. The proof is based on a truncated second moment method, which only requires to have good estimates on the quenched Green's function. |
| title | Recurrence and transience of the critical random walk snake in random conductances |
| topic | Probability 60K37, 60J80 (Primary) 82B41, 82D30 (Secondary) |
| url | https://arxiv.org/abs/2406.17622 |