Abstract
We endow the -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 . Previous work in this series used quantum Loewner evolution () to construct a metric on a countable dense subset of . Here, we show that a.s. extends uniquely and continuously to a metric on all of . Letting d denote the Euclidean metric on , we show that the identity map between and is a.s. Hölder continuous in both directions. We establish several other properties of , 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 balls and related quantum surfaces, as well as a careful analysis of geodesics.
Funding Statement
J.M.’s work was also partially supported by NSF Grant DMS-1204894 and J.M. thanks Institut Henri Poincaré for support as a holder of the Poincaré chair, during which part of this work was completed. S.S.’s work was also partially supported by NSF Grants DMS-1209044, DMS-1712862, a fellowship from the Simons Foundation and EPSRC Grants EP/L018896/1 and EP/I03372X/1.
Acknowledgments
We have benefited from conversations about this work with many people, a partial list of whom includes Omer Angel, Itai Benjamini, Nicolas Curien, Hugo Duminil-Copin, Amir Dembo, Bertrand Duplantier, Ewain Gwynne, Nina Holden, Jean-François Le Gall, Gregory Miermont, Rémi Rhodes, Steffen Rohde, Oded Schramm, Stanislav Smirnov, Xin Sun, Vincent Vargas, Menglu Wang, Samuel Watson, Wendelin Werner, David Wilson and Hao Wu.
We thank two anonymous referees for many helpful comments which have led to substantial improvements in the exposition.
We would also like to thank the Isaac Newton Institute (INI) for Mathematical Sciences, Cambridge, for its support and hospitality during the program on Random Geometry where part of this work was completed.
Citation
Jason Miller. Scott Sheffield. "Liouville quantum gravity and the Brownian map II: Geodesics and continuity of the embedding." Ann. Probab. 49 (6) 2732 - 2829, November 2021. https://doi.org/10.1214/21-AOP1506
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