A support theorem for exponential metrics of log-correlated Gaussian fields in arbitrary dimension

Abstract

Let $h$ be a log-correlated Gaussian field on ${\mathbb{R}}^d$, let $\gamma \in (0,\sqrt{2d}),$ let ${\mu }_h$ be the $\gamma$-Gaussian multiplicative chaos measure, and let $D_h$ be an exponential metric associated with $h$ satisfying certain natural axioms. In the special case when $d=2$, this corresponds to the Liouville quantum gravity (LQG) measure and metric. We show that the closed support of the law of $(D_h,{\mu }_h)$ includes all length metrics and probability measures on ${\mathbb{R}}^d$. That is, if $\mathfrak{d}$ is any length metric on ${\mathbb{R}}^d$ and $\mathfrak{m}$ is any probability measure on ${\mathbb{R}}^d$, then with positive probability $(D_h,{\mu }_h)$ is close to $(\mathfrak{d},\mathfrak{m})$ with respect to the uniform distance and the Prokhorov distance. Key ingredients include a scaling limit theorem for a first passage percolation type model associated with $h$, a special version of the white noise decomposition of $h$ in arbitrary dimension, and an approximation property by conformally flat Riemannian metrics in the uniform sense. Our results provide a robust tool to show that the LQG measure and metric, and its higher dimensional analogs, satisfy certain properties with positive probability.

Generalized result to higher dimensions, and split off appendix as a separate paper