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.
From: Zijie Zhuang [view email]
[v1]
Fri, 6 Oct 2023 03:51:38 UTC (593 KB)
[v2]
Thu, 2 Nov 2023 03:39:25 UTC (596 KB)
[v3]
Wed, 13 Mar 2024 01:57:21 UTC (809 KB)
[v4]
Mon, 7 Jul 2025 09:54:48 UTC (696 KB)
[v5]
Fri, 24 Jul 2026 16:05:07 UTC (691 KB)
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