Gaussian curvature on random planar maps and Liouville quantum gravity

TL;DR

This paper defines Gaussian curvature for Liouville quantum gravity surfaces and investigates its relation to discrete curvature on random planar maps, providing asymptotic results and convergence in distribution.

Contribution

It introduces a notion of Gaussian curvature for LQG surfaces and establishes its connection as a scaling limit of discrete curvature on random planar maps.

Findings

Discrete curvature integrated against test functions scales as ε^{o(1)}

Total discrete curvature on SLE segments converges to a random variable

Supports the conjecture that LQG curvature is the scaling limit of discrete curvature

Abstract

We investigate the notion of curvature in the context of Liouville quantum gravity (LQG) surfaces. We define the Gaussian curvature for LQG, which we conjecture is the scaling limit of discrete curvature on random planar maps. Motivated by this, we study asymptotics for the discrete curvature of $\epsilon$-mated CRT maps. More precisely, we prove that the discrete curvature integrated against a $C_c^2$ test function is of order $\epsilon^{o(1)},$ which is consistent with our scaling limit conjecture. On the other hand, we prove the total discrete curvature on a fixed space-filling SLE segment scaled by $\epsilon^{\frac{1}{4}}$ converges in distribution to an explicit random variable.