## Electronic Journal of Probability

### Coupling and an application to level-set percolation of the Gaussian free field

Alain-Sol Sznitman

#### Abstract

In the present article we consider a general enough set-up and obtain a refinement of the coupling between the Gaussian free field and random interlacements recently constructed by Titus Lupu in [9]. We apply our results to level-set percolation of the Gaussian free field on a $(d+1)$-regular tree, when $d \ge 2$, and derive bounds on the critical value $h_*$. In particular, we show that $0 < h_* < \sqrt{2u_*}$, where $u_*$ denotes the critical level for the percolation of the vacant set of random interlacements on a $(d+1)$-regular tree.

#### Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 35, 26 pp.

Dates
Accepted: 2 March 2016
First available in Project Euclid: 22 April 2016

https://projecteuclid.org/euclid.ejp/1461332876

Digital Object Identifier
doi:10.1214/16-EJP4563

Mathematical Reviews number (MathSciNet)
MR3492939

Zentralblatt MATH identifier
1336.60194

#### Citation

Sznitman, Alain-Sol. Coupling and an application to level-set percolation of the Gaussian free field. Electron. J. Probab. 21 (2016), paper no. 35, 26 pp. doi:10.1214/16-EJP4563. https://projecteuclid.org/euclid.ejp/1461332876

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