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

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.