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

*(English)*Zbl 1410.60037Summary: 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 \(\mathrm{CLE}_4\) 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 \(\mathrm{SLE}_4\) 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 \(\lambda \leq a + b < 4\lambda\) 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 \(\mathrm{CLE}_4\) 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.

Two-valued sets are the closure of the union of countably many \(\mathrm{SLE}_4\) 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 \(\lambda \leq a + b < 4\lambda\) 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 \(\mathrm{CLE}_4\) 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.

##### MSC:

60G15 | Gaussian processes |

60G60 | Random fields |

60D05 | Geometric probability and stochastic geometry |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

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
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\textit{J. Aru} and \textit{A. Sepúlveda}, Electron. J. Probab. 23, Paper No. 61, 35 p. (2018; Zbl 1410.60037)

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