Phase transition for the late points of random walk

• 成果类型: Article
• 署名作者: Prevost, Alexis; Rodriguez, Pierre-Francois; Sousi, Perla
• 署名单位: University of Bonn; Imperial College London; University of Cambridge
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-026-01490-0
• 发表日期: 2026-08
• 页码: 1189-1274
• 关键词: random interlacements; torus; times; set

摘要

Let X be a random walk on the torus of side length N in dimension d >= 3 with uniform starting point, and t(cov) be the expected value of its cover time, which is the first time that X has visited every vertex of the torus at least once. For alpha> 0, the set L-alpha of alpha-late points consists of those points not visited by X at time alpha t(cov). We prove the existence of a value alpha & lowast; is an element of ( 1/2 , 1) across which L-alpha trivialises as follows: for all alpha>alpha & lowast; and epsilon >= N-c there exists a coupling of L-alpha and two occupation sets B-alpha +/- of i.i.d. Bernoulli fields having the same density as L-alpha +/-epsilon, which is asymptotic to N-(alpha +/-epsilon)d , with the property that the inclusion B alpha+ subset of L-alpha subset of B alpha- holds with high probability as N ->infinity. On the contrary, when alpha <= alpha(& lowast;) there is no such coupling. Corresponding results also hold for the vacant set of random interlacements at high intensities. The transition at alpha(& lowast;) corresponds to the (dis-)appearance of 'double-points' (i.e. neighboring pairs of points) in L-alpha. We further describe the law of L-alpha for alpha> 1/2 by adding independent patterns to B-alpha +/- . In dimensions d >= 4 these are exactly all two-point sets. When d = 3 one must also include all connected three-point sets, but no other