Limit theorem for subdiffusive random walk in Dirichlet random environment in dimension $d \ge 3$
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911833138397184 |
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| author | Perrel, Adrien |
| author_facet | Perrel, Adrien |
| contents | We consider random walk in Dirichlet random environment in ${\mathbf{Z}^d, d\ge 3}$, which corresponds to the case where the environment is constructed from i.i.d. transition probabilities at each vertex with a Dirichlet distribution with parameters $(α_i)_{1 \le i \le 2d}$. Dirichlet environments are weakly elliptic and the walk can be slowdowned by local traps whose strength are governed by a parameter $κ$. In this paper we prove a stable limit theorem when the walk is ballistic but subdiffusive, i.e. when ${κ\in (1,2)}$. This completes the result of Poudevigne (arXiv:1909.03866) who proved a sub-ballistic stable limit theorem when $κ\in (0,1)$. Contrary to Poudevigne, we have to assume Sznitman's condition $\mathbf{(T)}$ to prove the limit theorem since we work at the level of fluctuations and need a better control on renewal times. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06502 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Limit theorem for subdiffusive random walk in Dirichlet random environment in dimension $d \ge 3$ Perrel, Adrien Probability We consider random walk in Dirichlet random environment in ${\mathbf{Z}^d, d\ge 3}$, which corresponds to the case where the environment is constructed from i.i.d. transition probabilities at each vertex with a Dirichlet distribution with parameters $(α_i)_{1 \le i \le 2d}$. Dirichlet environments are weakly elliptic and the walk can be slowdowned by local traps whose strength are governed by a parameter $κ$. In this paper we prove a stable limit theorem when the walk is ballistic but subdiffusive, i.e. when ${κ\in (1,2)}$. This completes the result of Poudevigne (arXiv:1909.03866) who proved a sub-ballistic stable limit theorem when $κ\in (0,1)$. Contrary to Poudevigne, we have to assume Sznitman's condition $\mathbf{(T)}$ to prove the limit theorem since we work at the level of fluctuations and need a better control on renewal times. |
| title | Limit theorem for subdiffusive random walk in Dirichlet random environment in dimension $d \ge 3$ |
| topic | Probability |
| url | https://arxiv.org/abs/2404.06502 |