The CRT is the scaling limit of random dissections |
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| Authors: | Nicolas Curien Bénédicte Haas Igor Kortchemski |
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| Institution: | 1. CNRS and Université Paris VI, Laboratoire de Probabilités et Modèles Aléatoires, France;2. Cérémade, Université Paris‐Dauphine, Place du Maréchal de Lattre de Tassigny 75016 Paris France, and Département de Mathématiques et Applications, école Normale Supérieure, France;3. Département de Mathématiques et Applications, école Normale Supérieure, France |
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| Abstract: | Abstract–We study the graph structure of large random dissections of polygons sampled according to Boltzmann weights, which encompasses the case of uniform dissections or uniform p‐angulations. As their number of vertices n goes to infinity, we show that these random graphs, rescaled by  , converge in the Gromov–Hausdorff sense towards a multiple of Aldous' Brownian tree when the weights decrease sufficiently fast. The scaling constant depends on the Boltzmann weights in a rather amusing and intriguing way, and is computed by making use of a Markov chain which compares the length of geodesics in dissections with the length of geodesics in their dual trees. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 304–327, 2015 |
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| Keywords: | random dissections Galton– Watson trees scaling limits Brownian continuum random tree Gromov– Hausdorff topology |
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