Tightness of exponential metrics for log-correlated Gaussian fields in arbitrary dimension

TL;DR

This paper establishes the tightness of a family of random metrics derived from log-correlated Gaussian fields in any dimension, advancing understanding of Liouville quantum gravity analogs.

Contribution

It proves the tightness and subsequential convergence of these metrics in arbitrary dimensions for the subcritical phase, extending previous results beyond two dimensions.

Findings

Metrics are tight after rescaling in all dimensions for subcritical parameters.

Every subsequential limit induces the Euclidean topology.

Provides a list of open problems for future research.

Abstract

We prove the tightness of a natural approximation scheme for an analog of the Liouville quantum gravity metric on $\mathbb R^d$ for arbitrary $d\geq 2$. More precisely, let $\{h_n\}_{n\geq 1}$ be a suitable sequence of Gaussian random functions which approximates a log-correlated Gaussian field on $\mathbb R^d$. Consider the family of random metrics on $\mathbb R^d$ obtained by weighting the lengths of paths by $e^{\xi h_n}$, where $\xi > 0$ is a parameter. We prove that if $\xi$ belongs to the subcritical phase (which is defined by the condition that the distance exponent $Q(\xi)$ is greater than $\sqrt{2d}$), then after appropriate re-scaling, these metrics are tight and that every subsequential limit is a metric on $\mathbb R^d$ which induces the Euclidean topology. We include a substantial list of open problems.