On bounded-type thin local sets of the two-dimensional Gaussian free field

TL;DR

This paper characterizes bounded-type thin local sets (BTLS) of the 2D Gaussian free field, linking them to CLE(4) structures, and constructs all BTLS with harmonic functions taking two values, showing they are measurable functions of the GFF.

Contribution

It provides a complete characterization of BTLS in relation to CLE(4) and constructs all such sets with two-valued harmonic functions, establishing their measurability.

Findings

BTLS are contained in nested CLE(4) carpets.

All BTLS are connected to the domain boundary.

BTLS with two-valued harmonic functions are measurable functions of the GFF.

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 CLE(4) 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 CLE(4) 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 CLE(4)) are in fact measurable functions of the GFF.