## The Annals of Applied Probability

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

#### Abstract

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

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

Dates
Revised: August 2015
First available in Project Euclid: 19 October 2016

https://projecteuclid.org/euclid.aoap/1476884307

Digital Object Identifier
doi:10.1214/15-AAP1165

Mathematical Reviews number (MathSciNet)
MR3563197

Zentralblatt MATH identifier
1353.60083

#### Citation

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. https://projecteuclid.org/euclid.aoap/1476884307

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