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

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 $3.4641\le d_{\sqrt{2}}\le 3.63299$ and in the limiting case we get $4.77485\le {\lim }_{\gamma \to 2^{-}}{d}_{\gamma }\le 4.89898$.

56 pages, 7 figues; final version, to appear in CMP

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• Duality of Chordal SLE
• Heat kernel for Liouville Brownian motion and Liouville graph distance
• Schnyder woods, SLE(16), and Liouville quantum gravity
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• Geodesic networks in Liouville quantum gravity surfaces
• Critical behaviour of loop models on causal triangulations
• Local convergence of large random triangulations coupled with an Ising model
• Plane bipolar orientations and quadrant walks
• Tightness of supercritical Liouville first passage percolation
• Uniqueness of the critical and supercritical Liouville quantum gravity metrics
• Planar maps and random partitions
• Box-counting dimension in one-dimensional random geometry of multiplicative cascades
• Regularity and confluence of geodesics for the supercritical Liouville quantum gravity metric
• Liouville metric of star-scale invariant fields: tails and Weyl scaling
• Weak Liouville quantum gravity metrics with matter central charge $\mathbf{c}\in (-\infty ,25)$
• Comparison of discrete and continuum Liouville first passage percolation
• Up-to-constants comparison of Liouville first passage percolation and Liouville quantum gravity
• Liouville quantum gravity metrics are not doubling
• Scale-invariant random geometry from mating of trees: a numerical study
• Introduction to the Liouville quantum gravity metric
• Volume of metric balls in Liouville quantum gravity
• On the chemical distance exponent for the two-sided level-set of the 2D Gaussian free field
• Environment seen from infinite geodesics in Liouville Quantum Gravity