Uniqueness of the critical and supercritical Liouville quantum gravity metrics

Abstract

We show that for each ${\mathbf{c}}_{\mathrm{M}}\in [1,25)$, there is a unique metric associated with Liouville quantum gravity (LQG) with matter central charge ${\mathbf{c}}_{\mathrm{M}}$. An earlier series of works by Ding-Dubédat-Dunlap-Falconet, Gwynne-Miller, and others showed that such a metric exists and is unique in the subcritical case ${\mathbf{c}}_{\mathrm{M}}\in (-\infty ,1)$, which corresponds to coupling constant $\gamma \in (0,2)$. The critical case ${\mathbf{c}}_{\mathrm{M}}=1$ corresponds to $\gamma =2$ and the supercritical case ${\mathbf{c}}_{\mathrm{M}}\in (1,25)$ corresponds to $\gamma \in \mathbb{C}$ with $|\gamma |=2$. Our metric is constructed as the limit of an approximation procedure called Liouville first passage percolation, which was previously shown to be tight for ${\mathbf{c}}_{\mathrm{M}}\in [1,25)$ by Ding and Gwynne (2020). In this paper, we show that the subsequential limit is uniquely characterized by a natural list of axioms. This extends the characterization of the LQG metric proven by Gwynne and Miller (2019) for ${\mathbf{c}}_{\mathrm{M}}\in (-\infty ,1)$ to the full parameter range ${\mathbf{c}}_{\mathrm{M}}\in (-\infty ,25)$. Our argument is substantially different from the proof of the characterization of the LQG metric for ${\mathbf{c}}_{\mathrm{M}}\in (-\infty ,1)$. In particular, the core part of the argument is simpler and does not use confluence of geodesics.

109 pages, 24 figures; to appear in Proceedings of the London Mathematical Society

References in corpus (2)

• The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds
• Geodesic networks in Liouville quantum gravity surfaces

Cited by in corpus (1)

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