On Bounded-Type Thin Local Sets of the Two-Dimensional Gaussian Free Field

Abstract

We study certain classes of local sets of the two-dimensional Gaussian free field (GFF) in a simply-connected domain, and their relation to the conformal loop ensemble CLE4 and its variants. More specifically, we consider bounded-type thin local sets (BTLS), where thin means that the local set is small in size, and bounded-type means that the harmonic function describing the mean value of the field away from the local set is bounded by some deterministic constant. We show that a local set is a BTLS if and only if it is contained in some nested version of the CLE4 carpet, and prove that all BTLS are necessarily connected to the boundary of the domain. We also construct all possible BTLS for which the corresponding harmonic function takes only two prescribed values and show that all these sets (and this includes the case of CLE4) are in fact measurable functions of the GFF.

The First Passage Sets of the 2D Gaussian Free Field: Convergence and Isomorphisms

“…Moreover, we also give an exact formula for the conditional probability that the two level lines agree in this coupling, conditioned on one of the level lines. In fact, in the non-boundary touching case, the existance of a coupling where level lines of two GFF-s with different boundary conditions agree with positive probability follows already from Proposition 13 in [ASW17]. Here, we provide an explicit such coupling with exact formulas.…”

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

“…One of the simplest families of BTLS are the generalised level lines, first described in [SS13]. We recall here some of their properties, see [WW16,ASW17] for a more thorough treatment of the subject. To simplify our statements take D := H. Furthermore, let u be a harmonic function in D. We say that (η(t)) t≥0 , a curve parametrised by half-plane capacity, is the generalised level line for the GFF Γ + u in D up to a stopping time τ if for all t ≥ 0: ( * ): The set η([0, t ∧ τ ]) is a BTLS of the GFF Γ, with harmonic function h t := h η([0,t∧τ ]) satisfying the following properties: h t + u is a harmonic function in D\η([0, min(t, τ )]) with boundary values −λ on the left-hand side of η, +λ on the right side of η, and with the same boundary values as u on ∂D.…”

“…Two-valued local sets (TVS) of the two-dimensional Gaussian free field (GFF), denoted A −a,b , were introduced in [ASW17]. They are the two-dimensional analogue of the exit times from an interval [−a, b] by a standard Brownian motion.…”

“…As the 2D GFF is not defined pointwise, one has to give meaning to TVS. This was done in [ASW17,ALS17], using the concept of local sets. This concept appeared first in the study of Markov random field in the 70s and 80s (see in particular [Roz82]) and it was reintroduced in [SS13] in the context of the coupling between the GFF and SLE 4 .…”

“…In [ASW17] two-valued sets were defined via their expected properties and their construction was provided using SLE 4 type of level lines. More precisely, it was shown that if a, b > 0, a + b ≥ 2λ and Γ is a GFF in a simply connected domain D, then there exists a unique local set A −a,b that satisfies…”