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dc.contributor.advisorScott Sheffield.en_US
dc.contributor.authorHolden, Nina, Ph. D. Massachusetts Institute of Technologyen_US
dc.contributor.otherMassachusetts Institute of Technology. Department of Mathematics.en_US
dc.date.accessioned2018-09-17T15:47:41Z
dc.date.available2018-09-17T15:47:41Z
dc.date.copyright2018en_US
dc.date.issued2018en_US
dc.identifier.urihttp://hdl.handle.net/1721.1/117865
dc.descriptionThesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018.en_US
dc.descriptionCataloged from PDF version of thesis.en_US
dc.descriptionIncludes bibliographical references (pages 247-259).en_US
dc.description.abstractThe Schramm-Loewner evolution (SLE) is a random fractal curve which describes the scaling limit of interfaces in a wide range of statistical physics models. Liouville quantum gravity (LQG) is a random fractal surface which arises as the scaling limit of discrete surfaces known as random planar maps (RPM). First, we study Hausdorff dimensions for SLE curves. We prove a KPZ-type formula which relates the Hausdorff dimension of an arbitrary subset of an SLE curve to the Hausdorff dimension of a time set for a two-dimensional correlated Brownian motion. Using our formula, we obtain new and simple proofs for a number of SLE Hausdorff dimensions, and we prove an explicit formula which says how much the Hausdorff dimension of a deterministic set increases upon being conformally mapped to an SLE curve. This is joint work with Gwynne and Miller. Then we introduce a mating-of-trees construction of SLE in Euclidean geometry in collaboration with Sun. This is the Euclidean counterpart to the mating-of-trees construction of SLE in an LQG environment by Duplantier, Miller, and Sheffield, which plays an essential role throughout the thesis. Finally, we prove scaling limit results for uniformly sampled RPM known as triangulations. In a joint work with Bernardi and Sun we show that a number of observables associated with critical site percolation on the triangulation converge jointly in law to the associated observables of SLE6 on an independent [square root of] 8/3-LQG surface. In a joint work with Sun we use this and other results to prove convergence of the triangulation under a discrete conformal embedding which we call the Cardy embedding. The conformally embedded triangulation induces an area measure and a metric on the complex plane, and we show that this measure and metric converge jointly in the scaling limit to an instance of the [square root of] 8/3-LQG disk (equivalently, to an instance of the conformally embedded Brownian disk).en_US
dc.description.statementofresponsibilityby Nina Holden.en_US
dc.format.extent259 pagesen_US
dc.language.isoengen_US
dc.publisherMassachusetts Institute of Technologyen_US
dc.rightsMIT 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.en_US
dc.rights.urihttp://dspace.mit.edu/handle/1721.1/7582en_US
dc.subjectMathematics.en_US
dc.titleCardy embedding of random planar maps and a KPZ formula for mated treesen_US
dc.typeThesisen_US
dc.description.degreePh. D.en_US
dc.contributor.departmentMassachusetts Institute of Technology. Department of Mathematics.en_US
dc.identifier.oclc1051190044en_US


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