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

2023/10/06 by Jian Ding, Ewain Gwynne, Ding, Jian +3 · 1 citation
#math.PR #math-ph #math.MP

paper · pdf · doi:10.48550/arxiv.2310.03996

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≥ 2. More precisely, let \h_n\_{n≥ 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^ξh_n, where ξ> 0 is a parameter. We prove that if ξ belongs to the subcritical phase (which is defined by the condition that the distance exponent Q(ξ) is greater than √(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.

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