Phase transition for the vacant set of random walk and random interlacements
arXiv (Cornell University)(2023)
摘要
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.
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关键词
random walk,phase transition,interlacements
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