Weak exponential metrics for high-dimensional log-correlated Gaussian fields

TL;DR

This paper introduces the concept of weak exponential metrics for high-dimensional log-correlated Gaussian fields, establishing their properties and extending Liouville quantum gravity metric theory beyond two dimensions.

Contribution

It defines weak $oldsymbol{ extgamma}$-exponential metrics for high-dimensional fields and proves their fundamental properties, extending existing 2D theories to higher dimensions.

Findings

Established existence of weak exponential metrics as subsequential limits.

Derived sharp moment bounds and optimal Hölder exponents.

Extended KPZ relation and Hausdorff dimension results to higher dimensions.

Abstract

For log-correlated Gaussian fields on $\mathbb{R}^d$ with $d \geq 2$, Ding-Gwynne-Zhuang (2023) established the existence of subsequential limits of exponential metrics obtained from appropriate approximations. For $\gamma \in (0,\sqrt{2d})$, we define a \textit{weak $\gamma$-exponential metric} to be a map $h \mapsto D_h$ that assigns to a sample of a log-correlated Gaussian field $h$ a continuous metric on $\mathbb{R}^d$ satisfying a list of axioms. We prove that every subsequential limit of exponential metrics built from appropriate approximations of $h$ is a weak $\gamma$-exponential metric in this sense. Moreover, we establish general properties that hold for any weak exponential metric: (1). sharp moment bounds for several natural distances; (2). optimal H\"older exponents when comparing $D_h$ and the Euclidean metric; and (3). Hausdorff dimension and a KPZ relation. These results extend the two-dimensional Liouville quantum gravity metric theory to higher dimensions. Along the way we derive several useful properties for log-correlated Gaussian fields including the equivalence between white-noise decomposition and convolution, and a shell independence lemma.