2023/05/24 by Andres A. Contreras Hip, Ewain Gwynne, Hip, Andres A. Contreras +1
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Noncommutative and Quantum Gravity Theories #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2305.15588
openalex publication_date 2023/05/24 · openalex created_date 2023/05/27 · openalex updated_date 2026/07/28
Let h be a log-correlated Gaussian field on \Rd, let γ∈ (0,√(2d)), let μh be the γ-Gaussian multiplicative chaos measure, and let Dh be an exponential metric associated with h satisfying certain natural axioms. In the special case when d=2, this corresponds to the Liouville quantum gravity (LQG) measure and metric. We show that the closed support of the law of (Dh,μh) includes all length metrics and probability measures on \Rd. That is, if \mathfrak d is any length metric on \Rd and \mathfrak m is any probability measure on \Rd, then with positive probability (Dh , μh) is close to (\mathfrak d , \mathfrak m) with respect to the uniform distance and the Prokhorov distance. Key ingredients include a scaling limit theorem for a first passage percolation type model associated with h, a special version of the white noise decomposition of h in arbitrary dimension, and an approximation property by conformally flat Riemannian metrics in the uniform sense. Our results provide a robust tool to show that the LQG measure and metric, and its higher dimensional analogs, satisfy certain properties with positive probability.