Liouville quantum gravity and the Brownian map I: the QLE (8 / 3 , 0) metric
Name
Miller-Sheffield2019_Article_LiouvilleQuantumGravityAndTheB.pdf
Description
Published version
Size
1.44 MB
Format
Adobe PDF
Checksum (MD5)
3f34f79fdd3226fb5c4090deb381323a
Author(s)
Sheffield, Scott Roger
Date Issued
July 2019
Journal
Inventiones mathematicae
Publisher
Springer Science and Business Media LLC
Citation
Miller, Jason and Scott Sheffield. “Liouville quantum gravity and the Brownian map I: the QLE (8 / 3 , 0) metric.” Inventiones mathematicae, vol. 219, 2019, pp. 75–152 © 2019 The Author(s)
Version
Final published version
Abstract
Liouville quantum gravity (LQG) and the Brownian map (TBM) are two distinct models o the LQG sphere comes equipped with a conformal structure, and TBM comes equipped with a metric space structure, and endowing either one with the other’s structure has been an open problem for some time. This paper is the first in a three-part series that unifies LQG and TBM by endowing each object with the other’s structure and showing that the resulting laws agree. The present work considers a growth process called quantum Loewner evolution (QLE) on a 8/3-LQG surface S and defines dQ(x, y) to be the amount of time it takes QLE to grow from x∈ S to y∈ S. We show that dQ(x, y) is a.s. determined by the triple (S, x, y) (which is far from clear from the definition of QLE) and that dQ a.s. satisfies symmetry (i.e., dQ(x, y) = dQ(y, x)) for a.a. (x, y) pairs and the triangle inequality for a.a. triples. This implies that dQ is a.s. a metric on any countable sequence sampled i.i.d. from the area measure on S. We establish several facts about the law of this metric, which are in agreement with similar facts known for TBM. The subsequent papers will show that this metric a.s. extends uniquely and continuously to the entire 8/3-LQG surface and that the resulting measure-endowed metric space is TBM.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution 4.0 International license
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s00222-019-00905-1
