The Annals of Applied Probability

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

Jiří Černý and Augusto Teixeira

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For $d\ge3$, we construct a new coupling of the trace left by a random walk on a large $d$-dimensional discrete torus with the random interlacements on $\mathbb{Z}^{d}$. This coupling has the advantage of working up to macroscopic subsets of the torus. As an application, we show a sharp phase transition for the diameter of the component of the vacant set on the torus containing a given point. The threshold where this phase transition takes place coincides with the critical value $u_{\star}(d)$ of random interlacements on $\mathbb{Z}^{d}$. Our main tool is a variant of the soft-local time coupling technique of Popov and Teixeira [J. Eur. Math. Soc. (JEMS) 17 (2015) 2545–2593].

Article information

Ann. Appl. Probab., Volume 26, Number 5 (2016), 2883-2914.

Received: December 2014
Revised: August 2015
First available in Project Euclid: 19 October 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60G50: Sums of independent random variables; random walks 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50] 05C80: Random graphs [See also 60B20]

Random walks percolation phase transition


Černý, Jiří; Teixeira, Augusto. Random walks on torus and random interlacements: Macroscopic coupling and phase transition. Ann. Appl. Probab. 26 (2016), no. 5, 2883--2914. doi:10.1214/15-AAP1165.

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