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

Abstract

We study two-valued local sets, $\\mathbb{A}_{-a,b}$, of the two-dimensional\ncontinuum Gaussian free field (GFF) with zero boundary condition in simply\nconnected domains. Intuitively, $\\mathbb{A}_{-a,b}$ is the (random) set of\npoints connected to the boundary by a path on which the values of the GFF\nremain in $[-a,b]$. For specific choices of the parameters $a, b$ the\ntwo-valued sets have the law of the CLE$_4$ carpet, the law of the union of\nlevel lines between all pairs of boundary points, or, conjecturally, the law of\nthe interfaces of the scaling limit of XOR-Ising model.\n Two-valued sets are the closure of the union of countably many SLE$_4$ type\nof loops, where each loop comes with a label equal to either $-a$ or $b$. One\nof the main results of this paper describes the connectivity properties of\nthese loops. Roughly, we show that all the loops are disjoint if $a+b \\geq\n4\\lambda$, and that their intersection graph is connected if $a + b <\n4\\lambda$. This also allows us to study the labels (the heights) of the loops.\nWe prove that the labels of the loops are a function of the set\n$\\mathbb{A}_{-a,b}$ if and only if $a\\neq b$ and $2\\lambda \\leq a+b < 4\\lambda$\nand that the labels are independent given the set if and only if $a = b =\n2\\lambda$. We also show that the threshold for the level-set percolation in the\n2D continuum GFF is $-2\\lambda$.\n Finally, we discuss the coupling of the labelled CLE$_4$ with the GFF. We\ncharacterise this coupling as a specific local set coupling, and show how to\napproximate these local sets. We further see how in these approximations the\nlabels naturally encode distances to the boundary.\n