Random walks on torus and random interlacements: Macroscopic coupling\n and phase transition

2014/11/28 by Jiří Černý, Černý, Jiří, Augusto Teixeira +1
Mathematics · Physics and Astronomy · #FOS: Mathematics #Probability (math.PR) #Quantum chaos and dynamical systems #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1411.7795

openalex publication_date 2014/11/28 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

For d\≥ 3 we construct a new coupling of the trace left by a random walk\non a large d-dimensional discrete torus with the random interlacements on\n mathbb Z^d. This coupling has the advantage of working up to macroscopic\nsubsets of the torus. As an application, we show a sharp phase transition for\nthe diameter of the component of the vacant set on the torus containing a given\npoint. The threshold where this phase transition takes place coincides with the\ncritical value u_*(d) of random interlacements on mathbb Z^d. Our main\ntool is a variant of the soft-local time coupling technique of [PT12].\n

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