Dimensions of Two-Valued Sets via Imaginary Chaos

Abstract

Abstract Two-valued sets are local sets of the 2D 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 ${\mathbb{A}}_{-a,b}$ equals $d=2-2\lambda ^2/(a+b)^2$. For the key two-point estimate needed to give the lower bound on dimension, 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 ${\mathbb{A}}_{-a,b}$ and discuss its relation with the $d$-dimensional conformal Minkowski content of ${\mathbb{A}}_{-a,b}$.