LIOUVILLE QUANTUM GRAVITY AND THE BROWNIAN MAP II: GEODESICS AND CONTINUITY OF THE EMBEDDING

• 成果类型: Article
• 署名作者: Miller, Jason; Sheffield, Scott
• 署名单位: University of Cambridge; Massachusetts Institute of Technology (MIT)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/21-AOP1506
• 发表日期: 2021
• 页码: 2732-2829
• 关键词: GAUSSIAN MULTIPLICATIVE CHAOS; kpz relation; heat kernel; sle; reversibility

摘要

We endow the root 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 S-2. Previous work in this series used quantum Loewner evolution (QLE) to construct a metric d(Q) on a countable dense subset of S-2. Here, we show that d(Q) a.s. extends uniquely and continuously to a metric (d) over bar (Q) on all of S-2. Letting d denote the Euclidean metric on S-2, we show that the identity map between (S-2, d) and (S-2, (d) over bar (Q)) is a.s. Holder continuous in both directions. We establish several other properties of (S-2, (d) over bar (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 (S-2, (d) over bar (Q)) geodesics.