Phase transition for the vacant set of random walk and random interlacements

2023/08/15 by Hugo Duminil‐Copin, Duminil-Copin, Hugo, Subhajit Goswami +7 · 3 citations
Mathematics · Physics and Astronomy · #05C81 #60G50 #60K35 #82B43 #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2308.07919

openalex publication_date 2023/08/15 · openalex created_date 2023/08/17 · openalex updated_date 2026/07/30

Abstract

We consider the set of points visited by the random walk on the discrete torus (ℤ/Nℤ)^d, for d ≥ 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 ℤ^d. Our findings also yield the analogous infinite-volume result, i.e. the long purported equality of three critical parameters u, u_* and u_** naturally associated to the vacant set of random interlacements.

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• On cable-graph percolation between dimensions 2 and 3
• Asymptotics for the percolation threshold of finitary random interlacements in four and higher dimensions
• Solidification estimates for random walks on supercritical percolation clusters

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