Matings of negatively correlated trees with applications to Schnyder woods and bipolar orientations
Name
1015183410-MIT.pdf
Description
Full printable version
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19.27 MB
Format
Adobe PDF
Checksum (MD5)
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Author(s)
Sun, Xin, Ph. D. Massachusetts Institute of Technology. Department of Mathematics
Advisor(s)
Scott R. Sheffield.
Date Issued
2017
Publisher
Massachusetts Institute of Technology
Abstract
In this thesis, we study the mating of trees approach to Liouville quantum gravity decorated with SLE curves and its application to the scaling limit theory of decorated random planar maps. We focus on the less investigated regime where the SLE parameter K is larger than 8. We obtain three main results. First, we identify the covariance of the Brownian motion in the Mating of Trees Theorem for K > 8, answering a question of Duplantier-Miller-Sheffield. Second, we prove the joint convergence of bipolar oriented triangulations and their dual in the peanosphere topology, confirming a conjecture of Kenyon-Miller-Sheffield-Wilson. Third, we prove the joint convergence of the three trees and their dual in a uniformly sampled Schnyder wood in the peanosphere topology. The third result also yields a description of the continuum limit of a widely used planar embedding due to Schnyder. The scaling limits in the second and third results involve Peano curves coupled in the same imaginary geometry with different angles. In order to establish the scaling limits, we extend the mating of trees theory to multiple Peano curves in the same imaginary geometry.
Description
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2017.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 233-244).
Subjects
Mathematics.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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