Dimension of two-valued sets via imaginary chaos

TL;DR

This paper determines the Hausdorff dimension of two-valued sets of the 2D Gaussian free field using imaginary chaos and constructs a measure supported on these sets, linking to conformal Minkowski content.

Contribution

It provides a precise formula for the Hausdorff dimension of two-valued sets and introduces a novel approach using imaginary chaos and vertex fields.

Findings

Hausdorff dimension of two-valued sets is 2-2λ²/(a+b)²

Constructs a non-trivial measure supported on these sets

Links the measure to conformal Minkowski content

Abstract

Two-valued sets are local sets of the two-dimensional Gaussian free field (GFF) that can be thought of as representing all points of the domain that may be connected to the boundary by a curve on which the GFF takes values only in [-a,b]. Two-valued sets exist whenever $a+b\geq 2\lambda$, where $\lambda$ depends explicitly on the normalization of the GFF. We prove that the almost sure Hausdorff dimension of the two-valued set $A_{-a,b}$ equals $d=2-2\lambda^2/(a+b)^2$. For the two-point estimate, we use the real part of a "vertex field" built from the purely imaginary Gaussian multiplicative chaos. We also construct a non-trivial $d$-dimensional measure supported on $A_{-a,b}$ and discuss its relation with the $d$-dimensional conformal Minkowski content for $A_{-a,b}$.