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

Ewain Gwynne,Jian Ding

doi:10.1007/s00220-019-03487-4

Abstract

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

Highlights

Our result for Liouville first passage percolation (Theorem 1.5) allows us to deduce that certain functions of γ and dγ are increasing in γ
This paper proves universality across different approximations of Liouville quantum gravity, it is known that the exponents associated with Liouville graph distance and the Liouville heat kernel are not universal among all log-correlated Gaussian free fields: see [DZ15,DZZ18b]
We provide a short heuristic argument for why one should expect the relationship between Liouville graph distance and Liouville first passage percolation exponents asserted in Theorem 1.5 (Sect. 2.3)

Summary

Introduction

We first show that the diameter (in the adjacency graph) of the set of -mass cells in the SLE/LQG representation of the mated-CRT map which intersect the Euclidean unit ball is of order −1/dγ +o (1) with high probability (Proposition 4.7), using the comparison results of the preceding subsection and the bounds for Liouville graph distance from Theorem 1.4. We use this to show that the volume of the graph distance ball of radius r in the mated-CRT map is of order r dγ +or (1) (essentially by taking = 1/r dγ ), and transfer to other planar maps using the coupling results of [GHS17]. The reader does not need any knowledge of this theory to understand Sect. 4 beyond the background we provide, so long as he or she is willing to take certain results as black boxes

Preliminaries

Estimates for Liouville Graph Distance and LFPP

S midpoints of the four corners of

Connection to random planar maps