The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds

TL;DR

This paper establishes the existence, universality, and bounds of the fractal dimension exponent in Liouville quantum gravity for all b3  (0,2), showing it varies continuously and increases with b3, with implications for random planar maps and related models.

Contribution

It proves the existence, continuity, and monotonicity of the fractal dimension exponent in b3-LQG and provides improved bounds for this exponent across the entire range.

Findings

The fractal dimension exponent d_b3 exists for all b3  (0,2).

d_b3 is a continuous, strictly increasing function of b3.

Explicit bounds for d_b3 are provided, notably for b3= and as b3 ightarrow 2^-.

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 the $\gamma$-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 $3.4641 \leq d_{\sqrt 2} \leq 3.63299$ and in the limiting case we get $4.77485 \leq \lim_{\gamma\rightarrow 2^-} d_\gamma \leq 4.89898$.