The Tutte Embedding of the Poisson–Voronoi Tessellation of the Brownian Disk Converges to √8/3-Liouville Quantum Gravity
Name
Gwynne2020_Article_TheTutteEmbeddingOfThePoissonV.pdf
Description
Published version
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1.1 MB
Format
Adobe PDF
Checksum (MD5)
c16a2d7a27da545296131b92e9a42a70
Author(s)
Gwynne, Ewain
Miller, Jason
Sheffield, Scott
Date Issued
2020
Journal
Communications in Mathematical Physics
Publisher
Springer Science and Business Media LLC
Version
Final published version
Abstract
© 2019, The Author(s). Recent works have shown that an instance of a Brownian surface (such as the Brownian map or Brownian disk) a.s. has a canonical conformal structure under which it is equivalent to a 8/3-Liouville quantum gravity (LQG) surface. In particular, Brownian motion on a Brownian surface is well-defined. The construction in these works is indirect, however, and leaves open a basic question: is Brownian motion on a Brownian surface the limit of simple random walk on increasingly fine discretizations of that surface, the way Brownian motion on R2 is the ϵ→ 0 limit of simple random walk on ϵZ2? We answer this question affirmatively by showing that Brownian motion on a Brownian surface is (up to time change) the λ→ ∞ limit of simple random walk on the Voronoi tessellation induced by a Poisson point process whose intensity is λ times the associated area measure. Among other things, this implies that as λ→ ∞ the Tutte embedding (a.k.a. harmonic embedding) of the discretized Brownian disk converges to the canonical conformal embedding of the continuum Brownian disk, which in turn corresponds to 8/3-LQG. Along the way, we obtain other independently interesting facts about conformal embeddings of Brownian surfaces, including information about the Euclidean shapes of embedded metric balls and Voronoi cells. For example, we derive moment estimates that imply, in a certain precise sense, that these shapes are unlikely to be very long and thin.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution 4.0 International license
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DOI of Published Version
https://doi.org/10.1007/S00220-019-03610-5
