## Electronic Journal of Probability

### Two-valued local sets of the 2D continuum Gaussian free field: connectivity, labels, and induced metrics

#### Abstract

We study two-valued local sets, $\mathbb{A} _{-a,b}$, of the two-dimensional continuum Gaussian free field (GFF) with zero boundary condition in simply connected domains. Intuitively, $\mathbb{A} _{-a,b}$ is the (random) set of points connected to the boundary by a path on which the values of the GFF remain in $[-a,b]$. For specific choices of the parameters $a, b$ the two-valued sets have the law of the CLE$_4$ carpet, the law of the union of level lines between all pairs of boundary points, or, conjecturally, the law of the interfaces of the scaling limit of XOR-Ising model.

Two-valued sets are the closure of the union of countably many SLE$_4$ type of loops, where each loop comes with a label equal to either $-a$ or $b$. One of the main results of this paper describes the connectivity properties of these loops. Roughly, we show that all the loops are disjoint if $a+b \geq 4\lambda$, and that their intersection graph is connected if $a + b < 4\lambda$. This also allows us to study the labels (the heights) of the loops. We prove that the labels of the loops are a function of the set $\mathbb{A} _{-a,b}$ if and only if $a\neq b$ and $2\lambda \leq a+b < 4\lambda$ and that the labels are independent given the set if and only if $a = b = 2\lambda$. We also show that the threshold for the level-set percolation in the 2D continuum GFF is $-2\lambda$.

Finally, we discuss the coupling of the labelled CLE$_4$ with the GFF. We characterise this coupling as a specific local set coupling, and show how to approximate these local sets. We further see how in these approximations the labels naturally encode distances to the boundary.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 61, 35 pp.

Dates
Accepted: 29 May 2018
First available in Project Euclid: 20 June 2018

https://projecteuclid.org/euclid.ejp/1529460159

Digital Object Identifier
doi:10.1214/18-EJP182

Mathematical Reviews number (MathSciNet)
MR3827968

Zentralblatt MATH identifier
06924673

#### Citation

Aru, Juhan; Sepúlveda, Avelio. Two-valued local sets of the 2D continuum Gaussian free field: connectivity, labels, and induced metrics. Electron. J. Probab. 23 (2018), paper no. 61, 35 pp. doi:10.1214/18-EJP182. https://projecteuclid.org/euclid.ejp/1529460159

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