## Journal of the Mathematical Society of Japan

### Disconnection and level-set percolation for the Gaussian free field

Alain-Sol SZNITMAN

#### Abstract

We study the level-set percolation of the Gaussian free field on $\mathbb Z^d$, $d \ge 3$. We consider a level $\alpha$ such that the excursion-set of the Gaussian free field above $\alpha$ percolates. We derive large deviation estimates on the probability that the excursion-set of the Gaussian free field below the level $\alpha$ disconnects a box of large side-length from the boundary of a larger homothetic box. It remains an open question whether our asymptotic upper and lower bounds are matching. With the help of a recent work of Lupu [21], we are able to infer some asymptotic upper bounds for similar disconnection problems by random interlacements, or by simple random walk.

#### Article information

Source
J. Math. Soc. Japan, Volume 67, Number 4 (2015), 1801-1843.

Dates
First available in Project Euclid: 27 October 2015

https://projecteuclid.org/euclid.jmsj/1445951168

Digital Object Identifier
doi:10.2969/jmsj/06741801

Mathematical Reviews number (MathSciNet)
MR3417515

Zentralblatt MATH identifier
1337.60246

#### Citation

SZNITMAN, Alain-Sol. Disconnection and level-set percolation for the Gaussian free field. J. Math. Soc. Japan 67 (2015), no. 4, 1801--1843. doi:10.2969/jmsj/06741801. https://projecteuclid.org/euclid.jmsj/1445951168

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