Matings of negatively correlated trees with applications to Schnyder woods and bipolar orientations

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.

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2017

Author dc:creator

• Sun, Xin, Ph. D. Massachusetts Institute of Technology. Department of Mathematics

Advisor dc:contributor.advisor

• Scott R. Sheffield.

• Mathematics.

Statement dc:rights

• MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/112892

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/112892