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

Abstract

Let $h$ be a log-correlated Gaussian field on $\R^d$, let $γ\in (0,\sqrt{2d}),$ let $μ_h$ be the $γ$-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 $(D_h,μ_h)$ includes all length metrics and probability measures on $\R^d$. That is, if $\mathfrak d$ is any length metric on $\R^d$ and $\mathfrak m$ is any probability measure on $\R^d$, then with positive probability $(D_h , μ_h)$ 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.