2026-09-19T12:06:30Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/1178712026-06-16T18:55:07Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Scott R. Sheffield.
Gwynne, Ewain
Massachusetts Institute of Technology. Department of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2018-09-17T15:47:56Z
2018-09-17T15:47:56Z
2018
2018
http://hdl.handle.net/1721.1/117871
1051190321
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 457-470).
A random planar map is a graph embedded in the sphere, viewed modulo orientation-preserving homeomorphisms. Random planar maps are the discrete analogues of random fractal surfaces called [gamma]-Liouville quantum gravity (LQG) surfaces with parameter [gamma] E (0, 2]. We study the large-scale structure of random planar maps (and statistical mechanics models on them) viewed as metric measure spaces equipped with the graph distance and the counting measure on vertices. In particular, we show that uniform random planar maps (which correspond to the case [gamma]= [square root of]8/3) decorated by a self-avoiding walk or a critical percolation interface converge in the scaling limit to [square root of]8/3- LQG surfaces decorated by SLE8/3 and SLE6, respectively, with respect to a generalization of the Gromov-Hausdorff topology. We also introduce an approach for analyzing certain random planar maps belonging to the [gamma]-LQG universality class for general [gamma] E (0, 2) and use this approach to prove several estimates for graph distances in such maps.
by Ewain Gwynne.
Ph.D.
470 pages
eng
Massachusetts Institute of Technology
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.
http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
On the metric structure of random planar maps and SLE-decorated Liouville quantum gravity
Thesis
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<Abstract>A random planar map is a graph embedded in the sphere, viewed modulo orientation-preserving homeomorphisms. Random planar maps are the discrete analogues of random fractal surfaces called [gamma]-Liouville quantum gravity (LQG) surfaces with parameter [gamma] E (0, 2]. We study the large-scale structure of random planar maps (and statistical mechanics models on them) viewed as metric measure spaces equipped with the graph distance and the counting measure on vertices. In particular, we show that uniform random planar maps (which correspond to the case [gamma]= [square root of]8/3) decorated by a self-avoiding walk or a critical percolation interface converge in the scaling limit to [square root of]8/3- LQG surfaces decorated by SLE8/3 and SLE6, respectively, with respect to a generalization of the Gromov-Hausdorff topology. We also introduce an approach for analyzing certain random planar maps belonging to the [gamma]-LQG universality class for general [gamma] E (0, 2) and use this approach to prove several estimates for graph distances in such maps.</Abstract>
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