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

• Main Authors: Ding, Jian, Gwynne, Ewain, Zhuang, Zijie
• Format: Journal Article
• Language: English
• Published: 05.10.2023
• Subjects: Mathematics - Mathematical Physics; Mathematics - Probability; Physics - Mathematical Physics
• DOI: 10.48550/arxiv.2310.03996

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.