Abstract
Previous works in this series have shown that an instance of a sqrt{8/3}-Liouville quantum gravity (LQG) sphere has a well-defined distance function, and that the resulting metric measure space (mm-space) agrees in law with the Brownian map (TBM). In this work, we show that given just the mm-space structure, one can a.s. recover the LQG sphere. This implies that there is a canonical way to parameterize an instance of TBM by the Euclidean sphere (up to Möbius transformation). In other words, an instance of TBM has a canonical conformal structure. The conclusion is that TBM and the sqrt{8/3}-LQG sphere are equivalent. They ultimately encode the same structure (a topological sphere with a measure, a metric, and a conformal structure) and have the same law. From this point of view, the fact that the conformal structure a.s. determines the metric and vice-versa can be understood as a property of this unified law. The results of this work also imply that the analogous facts hold for Brownian and sqrt{8/3}-LQG surfaces with other topologies.
Highlights
The unit area Brownian map (T√BM) and the unit area Liouville quantum gravity (LQG) sphere are two natural continuum models for “random surfaces” which are both homeomorphic to the Euclidean sphere S2
An instance of the Brownian map (TBM) comes endowed with a metric space structure while an instance of the LQG sphere comes endowed with a conformal structure
The current paper shows that the second object a.s. determines the first; given only SBM one can a.s. reconstruct both S and φ. This implies that there is a.s. a canonical way to endow an instance of TBM with a conformal structure
Summary
The current paper shows that the second object a.s. determines the first; given only SBM one can a.s. reconstruct both S and φ This implies that there is a.s. a canonical way to endow an instance of TBM with a conformal structure (namely, the conformal structure it inherits from S). This paper will show that the method for recovering the conformal structure from the metric is local in a similar sense These statements imply tha√t we can impose a canonical length space structure on any surface that looks like a 8/3-LQG sphere locally, and a canonical conformal structure on any surface that looks locally like an instance of TBM. These variants can be defined as random variables on the same space—the space of (possibly marked) conformal mm-spaces—and that comparing the laws of surfaces defined on the LQG side with the laws of surfaces defined on the Brownian map side is at least possible in principle
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