Phase transition for the vacant set of random walk and random interlacements
| Autor: | Duminil-Copin, Hugo, Goswami, Subhajit, Rodriguez, Pierre-François, Severo, Franco, Teixeira, Augusto |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We consider the set of points visited by the random walk on the discrete torus $(\mathbb{Z}/N\mathbb{Z})^d$, for $d \geq 3$, at times of order $uN^d$, for a parameter $u>0$ in the large-$N$ limit. We prove that the vacant set left by the walk undergoes a phase transition across a non-degenerate critical value $u_* = u_*(d)$, as follows. For all $u< u_*$, the vacant set contains a giant connected component with high probability, which has a non-vanishing asymptotic density and satisfies a certain local uniqueness property. In stark contrast, for all $u> u_*$ the vacant set scatters into tiny connected components. Our results further imply that the threshold $u_*$ precisely equals the critical value, introduced by Sznitman in arXiv:0704.2560, which characterizes the percolation transition of the corresponding local limit, the vacant set of random interlacements on $\mathbb{Z}^d$. Our findings also yield the analogous infinite-volume result, i.e. the long purported equality of three critical parameters $\bar u$, $u_*$ and $u_{**}$ naturally associated to the vacant set of random interlacements. Comment: 94 pages, 2 figures |
| Databáze: | arXiv |
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