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Two-valued local sets of the 2D continuum Gaussian free field: connectivity, labels, and induced metrics. (English) Zbl 1410.60037
Summary: 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.

##### 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)
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