Electronic Journal of Probability

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

Juhan Aru and Avelio Sepúlveda

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1529460159

Digital Object Identifier
doi:10.1214/18-EJP182

Mathematical Reviews number (MathSciNet)
MR3827968

Zentralblatt MATH identifier
06924673

Subjects
Primary: 60G15: Gaussian processes 60G60: Random fields 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G60: Random fields 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J67: Stochastic (Schramm-)Loewner evolution (SLE)

Keywords
Gaussian free field local sets two-valued local sets conformal loop ensemble Schramm-Loewner evolution level lines level set percolation Lévy transform XOR-Ising

Rights
Creative Commons Attribution 4.0 International License.

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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References

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