{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/112892"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/112892","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Matings of negatively correlated trees with applications to Schnyder woods and bipolar orientations","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Sun, Xin, Ph. D. Massachusetts Institute of Technology. Department of Mathematics"],"institution":"Massachusetts Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Department of Mathematics.","school":null,"contributors":[],"advisors":["Scott R. Sheffield."],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017","date_published":"2017","updated_at":"2026-07-22T22:20:51Z","subjects":["Mathematics."],"languages":["eng"],"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."],"rights_urls":["http://dspace.mit.edu/handle/1721.1/7582"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1721.1/112892","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Scott R. Sheffield."]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. Department of Mathematics."]},{"key":"dc:contributor.other","label":"Dc Contributor Other","values":["Massachusetts Institute of Technology. Department of Mathematics."]},{"key":"dc:creator","label":"Author","values":["Sun, Xin, Ph. D. Massachusetts Institute of Technology. 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They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission."]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://dspace.mit.edu/handle/1721.1/7582"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1721.1/112892"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2017.","Cataloged from PDF version of thesis.","Includes bibliographical references (pages 233-244)."]},{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:title","label":"Title","values":["Matings of negatively correlated trees with applications to Schnyder woods and bipolar orientations"]}]}],"canonical_facts":{"dc:contributor.advisor":["Scott R. Sheffield."],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Mathematics."],"dc:contributor.other":["Massachusetts Institute of Technology. Department of Mathematics."],"dc:creator":["Sun, Xin, Ph. D. Massachusetts Institute of Technology. Department of Mathematics"],"dc:date.accessioned":["2017-12-20T18:16:19Z"],"dc:date.available":["2017-12-20T18:16:19Z"],"dc:date.issued":["2017"],"dc:description":["Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2017.","Cataloged from PDF version of thesis.","Includes bibliographical references (pages 233-244)."],"dc:description.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."],"dc:description.degree":["Ph. D."],"dc:identifier.uri":["http://hdl.handle.net/1721.1/112892"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"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."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Mathematics."],"dc:title":["Matings of negatively correlated trees with applications to Schnyder woods and bipolar orientations"],"dc:type":["Thesis"]},"updated_at":"2026-07-22T22:20:51Z"}