Coupling the Gaussian Free Fields with Free and with Zero Boundary Conditions via Common Level Lines

Abstract

We point out a new simple way to couple the Gaussian Free Field (GFF) with free boundary conditions in a two-dimensional domain with the GFF with zero boundary conditions in the same domain: Starting from the latter, one just has to sample at random all the signs of the height gaps on its boundary-touching zero-level lines (these signs are alternating for the zeroboundary GFF) in order to obtain a free boundary GFF. Constructions and couplings of the free boundary GFF and its level lines via soups of reflected Brownian loops and their clusters are also discussed. Such considerations show for instance that in a domain with an axis of symmetry, if one looks at the overlay of a single usual Conformal Loop Ensemble CLE3 with its own symmetric image, one obtains the CLE4-type collection of level lines of a GFF with mixed zero/free boundary conditions in the half-domain.

Liouville measure as a multiplicative cascade via level sets of the Gaussian free field

“…Thus, almost surely the measures M γ a converge weakly to υ γ . It has recently been proven in [26] that sets satisfying the above hypothesis do exist, and that they can be used to couple the Dirichlet GFF with the Neumman GFF. Let us describe some concrete examples of these sets.…”

“…Let us describe some concrete examples of these sets. If Γ is a Dirichlet-Neumann GFF, then in [26] it is shown that there exists a (measurable) thin local set A(Γ) of the GFF such that:…”

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

“…TVS were defined in [ASW17] as the equivalent of the exit times of an interval by a one-dimensional Brownian motion. After that, they have been used as a tool to construct the Liouville Quantum gravity [APS17], to couple the Dirichlet and the free boundary GFF [QW17], and to study first passage sets [ALS17].…”

“…ALEs are used as a building block to construct more general TVS and are central in their study. Also, they were used in [QW17] to couple the Dirichlet and Neumann free fields.…”

“…Intuitively, they correspond to the set of points of the domain that can be connected to the boundary via a path on which the GFF takes values only in [−a, b]. TVS are tightly linked to the study of the 2D GFF: for λ = π/8 the set A −2λ,2λ describes the outer boundaries of the outermost sign clusters of the 2D GFF [Lup15,QW15], and Qian and Werner used A −λ,λ to couple the free boundary and zero boundary GFFs [QW17]. TVS also appear naturally in other statistical physics models: for example it is known that CLE 4 has the law of A −2λ,2λ [MS,ASW17] and moreover it is conjectured that A −a,b with a + b = 2(1 + √ 2)λ should be the scaling limit of interfaces corresponding to the XOR-Ising model [Wil11].…”