First passage percolation, local uniqueness for interlacements and capacity of random walk
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914091600183296 |
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| author | Prévost, Alexis |
| author_facet | Prévost, Alexis |
| contents | The study of first passage percolation (FPP) for the random interlacements model has been initiated in arXiv:2112.12096, where it is shown that on $\mathbb{Z}^d$, $d\geq 3$, the FPP distance is comparable to the graph distance with high probability. In this article, we give an asymptotically sharp lower bound on this last probability, which additionally holds on a large class of transient graphs with polynomial volume growth and polynomial decay of the Green function. When considering the interlacement set in the low-intensity regime, the previous bound is in fact valid throughout the near-critical phase. In low dimension, we also present two applications of this FPP result: sharp large deviation bounds on local uniqueness of random interlacements, and on the capacity of a random walk in a ball. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_03880 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | First passage percolation, local uniqueness for interlacements and capacity of random walk Prévost, Alexis Probability Mathematical Physics 60K35, 82B43, 60G50, 05C81 The study of first passage percolation (FPP) for the random interlacements model has been initiated in arXiv:2112.12096, where it is shown that on $\mathbb{Z}^d$, $d\geq 3$, the FPP distance is comparable to the graph distance with high probability. In this article, we give an asymptotically sharp lower bound on this last probability, which additionally holds on a large class of transient graphs with polynomial volume growth and polynomial decay of the Green function. When considering the interlacement set in the low-intensity regime, the previous bound is in fact valid throughout the near-critical phase. In low dimension, we also present two applications of this FPP result: sharp large deviation bounds on local uniqueness of random interlacements, and on the capacity of a random walk in a ball. |
| title | First passage percolation, local uniqueness for interlacements and capacity of random walk |
| topic | Probability Mathematical Physics 60K35, 82B43, 60G50, 05C81 |
| url | https://arxiv.org/abs/2309.03880 |