The Fractal Dimension of Liouville Quantum Gravity: Universality, Monotonicity, and Bounds

Abstract

Abstract: We prove that for each γ∈(0, 2), there is an exponent dγ>2, the “fractal dimension of γ-Liouville quantum gravity (LQG)”, which describes the ball volume growth exponent for certain random planar maps in theγ-LQG universality class, the exponent for the Liouville heat kernel, and exponents for various continuum approximations of γ-LQG distances such as Liouville graph distance and Liouville first passage percolation. We also show that dγ is a continuous, strictly increasing function of γ and prove upper and lower bounds for dγ which in some cases greatly improve on previously known bounds for the aforementioned exponents. For example, for γ=2 (which corresponds to spanning-tree weighted planar maps) our bounds give 3.4641≤d2≤3.63299 and in the limiting case we get 4.77485≤limγ→2-dγ≤4.89898.