[Paper Review] Liouville quantum gravity and the Brownian map II: geodesics and continuity of the embedding

Jason Miller, Scott Sheffield⋆|arXiv (Cornell University)|May 11, 2016

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper constructs a canonical metric on the $\sqrt{8/3}$-Liouville quantum gravity (LQG) sphere by extending a distance function defined on a dense set via quantum Loewner evolution (QLE), proving the resulting metric space is almost surely equivalent in law to the Brownian map. It establishes Hölder continuity of the identity map between the Euclidean and QLE metrics, and extends the result to the Brownian disk and plane, completing the equivalence between $\sqrt{8/3}$-LQG and the Brownian map as random metric measure spaces.

ABSTRACT

We endow the $\sqrt{8/3}$-Liouville quantum gravity sphere with a metric space structure and show that the resulting metric measure space agrees in law with the Brownian map. Recall that a Liouville quantum gravity sphere is a priori naturally parameterized by the Euclidean sphere ${\mathbf S}^2$. Previous work in this series used quantum Loewner evolution (QLE) to construct a metric $d_{\mathcal Q}$ on a countable dense subset of ${\mathbf S}^2$. Here we show that $d_{\mathcal Q}$ a.s. extends uniquely and continuously to a metric $\bar{d}_{\mathcal Q}$ on all of ${\mathbf S}^2$. Letting $d$ denote the Euclidean metric on ${\mathbf S}^2$, we show that the identity map between $({\mathbf S}^2, d)$ and $({\mathbf S}^2, \bar{d}_{\mathcal Q})$ is a.s. Hölder continuous in both directions. We establish several other properties of $({\mathbf S}^2, \bar{d}_{\mathcal Q})$, culminating in the fact that (as a random metric measure space) it agrees in law with the Brownian map. We establish analogous results for the Brownian disk and plane. Our proofs involve new estimates on the size and shape of QLE balls and related quantum surfaces, as well as a careful analysis of $({\mathbf S}^2, \bar{d}_{\mathcal Q})$ geodesics.

Motivation & Objective

• To construct a canonical metric on the $\sqrt{8/3}$-LQG sphere using quantum Loewner evolution (QLE) on a dense set of points.
• To show that the QLE distance function extends uniquely and continuously to the entire sphere, yielding a well-defined metric space structure.
• To prove that the resulting metric measure space is almost surely equivalent in law to the Brownian map.
• To extend the equivalence to the Brownian disk and plane via analogous constructions on quantum disks and cones.
• To establish Hölder continuity of the identity map between the Euclidean and QLE metrics on the sphere.

Proposed method

• Use QLE(8/3,0) on a $\sqrt{8/3}$-quantum cone to define a distance $d_{\mathcal{Q}}$ on a countable dense subset of the sphere.
• Apply quantitative Kolmogorov–Čentsov criteria to prove almost sure uniform continuity of $d_{\mathcal{Q}}$, enabling unique extension to $\overline{d}_{\mathcal{Q}}$ on the full sphere.
• Establish moment bounds and tail estimates for QLE hulls to control the size and shape of quantum surfaces and their metric balls.
• Use reverse exploration techniques and time-reversal of $\mathrm{SLE}_6$ to analyze the geometry of unexplored domains and metric bands.
• Construct approximations to geodesics via the $3/2$-Lévy net, a random spatial branching structure derived from QLE growth.
• Analyze the interplay between quantum area, boundary length, and Euclidean size to derive Hölder continuity of the identity map between $({\mathbf{S}}^2, d)$ and $({\mathbf{S}}^2, \overline{d}_{\mathcal{Q}})$.

Experimental results

Research questions

• RQ1 Can the QLE distance function on a dense subset of the $\sqrt{8/3}$-LQG sphere be uniquely and continuously extended to the entire sphere?
• RQ2 Is the resulting metric space structure on the $\sqrt{8/3}$-LQG sphere almost surely equivalent in law to the Brownian map?
• RQ3 Is the identity map between the Euclidean and QLE metrics on the sphere almost surely Hölder continuous in both directions?
• RQ4 Can analogous results be established for the Brownian disk and plane via quantum disks and cones?
• RQ5 Does the QLE-based metric on $\sqrt{8/3}$-LQG surfaces induce a canonical conformal structure on the Brownian map?

Key findings

• The QLE distance function $d_{\mathcal{Q}}$ on a dense subset of the $\sqrt{8/3}$-LQG sphere extends uniquely and continuously to a metric $\overline{d}_{\mathcal{Q}}$ on the entire sphere almost surely.
• The identity map from $({\mathbf{S}}^2, d)$ to $({\mathbf{S}}^2, \overline{d}_{\mathcal{Q}})$ is almost surely Hölder continuous in both directions.
• The random metric measure space $({\mathbf{S}}^2, \overline{d}_{\mathcal{Q}}, \mu)$ agrees in law with the Brownian map, where $\mu$ is the area measure.
• Analogous results hold for the Brownian disk and plane, with the quantum disk and cone providing the underlying $\sqrt{8/3}$-LQG surfaces.
• The construction yields a canonical metric on any $\sqrt{8/3}$-LQG surface, including the torus, and implies the Brownian map admits a unique conformal structure a.s.
• The results support the conjecture that the heat kernel for Liouville Brownian motion satisfies a $t^{-1} \exp(-d^{4/3}/t^{1/3})$ scaling, consistent with the QLE metric.