Translator Disclaimer
2018 Two-valued local sets of the 2D continuum Gaussian free field: connectivity, labels, and induced metrics
Juhan Aru, Avelio Sepúlveda
Electron. J. Probab. 23: 1-35 (2018). DOI: 10.1214/18-EJP182


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.


Download Citation

Juhan Aru. Avelio Sepúlveda. "Two-valued local sets of the 2D continuum Gaussian free field: connectivity, labels, and induced metrics." Electron. J. Probab. 23 1 - 35, 2018.


Received: 19 January 2018; Accepted: 29 May 2018; Published: 2018
First available in Project Euclid: 20 June 2018

zbMATH: 06924673
MathSciNet: MR3827968
Digital Object Identifier: 10.1214/18-EJP182

Primary: 60D05, 60G15, 60G60
Secondary: 60G60, 60J67, 60K35


Vol.23 • 2018
Back to Top