Weak Liouville quantum gravity metrics with matter central charge $\mathbf{c} \in (-\infty, 25)$

TL;DR

This paper constructs and analyzes a new class of Liouville quantum gravity metrics for all matter central charges less than 25, revealing novel geometric properties and extending the theory beyond the previously understood regime.

Contribution

It defines a rigorous LQG metric for all $ extbf{c} < 25$ using axioms and approximations, including the critical case $ extbf{c} = 1$, and explores their geometric and probabilistic properties.

Findings

Metrics are dense with singular points for $ extbf{c} ext{ in } (1,25)$

Metric balls are non-compact with infinite Hausdorff boundary dimension

Many properties from $ extbf{c} < 1$ extend to the full range $(- olinebreak extbf{infty}, 25)$

Abstract

Physics considerations suggest that a theory of Liouville quantum gravity (LQG) should exist for all values of matter central charge $\mathbf{c} \in (-\infty,25)$. Probabilists have rigorously defined LQG as a random metric measure space for $\mathbf{c} < 1$; however, they have only very recently begun to explore LQG in the $\mathbf{c} \in (1,25)$ phase. We define a random metric associated to LQG for all $\mathbf{c} < 25$ by a collection of axioms analogous to axioms stated in the $\mathbf{c} < 1$ setting. We show that such a metric exists for each $\mathbf{c}$ by considering an approximating metric known as Liouville first passage percolation. Ding and Gwynne proved that these approximating metrics are tight in a suitably chosen topology; we show that every subsequential limit satisfies our axioms. In particular, our result defines a metric associated to LQG in the critical case $\mathbf{c} =1$. The metrics we define for $\mathbf{c} \in (1,25)$ exhibit geometric behavior that sharply contrasts with the $\mathbf{c} < 1$ regime. We show that, for $\mathbf{c} \in (1,25)$, the metrics do not induce the Euclidean topology since they a.s. have a dense (measure zero) set of singular points, points at infinite distance from all other points. We use this fact to prove that a.s. the metric ball is not compact and its boundary has infinite Hausdorff dimension. Despite these differences, we demonstrate that many properties of LQG metrics for $\mathbf{c} < 1$ extend in some form to the entire range $(-\infty, 25)$. We show that the metrics are a.s. reverse H\"older continuous with respect to the Euclidean metric, are a.s. complete and geodesic away from the set of singular points, and satisfy the bounds for set-to-set distances that hold in the $\mathbf{c} < 1$ phase. Finally, we prove that the metrics satisfy a version of the (geometric) Knizhnik-Polyakov-Zamolodchikov (KPZ) formula.