Work

Gaussian curvature on random planar maps and Liouville quantum gravity

Year: 2025

Type: article

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 $$\epsilon $$ ϵ -mated CRT maps. More precisely, we prove that the discrete curvature integrated against a $$C_c^2$$ C c 2 test function is of order $$\epsilon ^{o(1)},$$ ϵ 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 $$\epsilon ^{\frac{1}{4}}$$ ϵ 1 4 converges in distribution to an explicit random variable.

Source: Probability Theory and Related Fields

DOI: 10.1007/s00440-025-01430-4

Authors: Andres A. Contreras Hip, Ewain Gwynne

Institution: University of Chicago

Language: English

Topic: Stochastic processes and statistical mechanics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090

Funder: National Science Foundation