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The CRT is the scaling limit of random dissections

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 $n^{-1/2}$, 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.
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Contributor : Bénédicte Haas <>
Submitted on : Thursday, May 16, 2013 - 1:04:34 PM
Last modification on : Friday, June 12, 2020 - 11:02:05 AM

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  • HAL Id : hal-00823219, version 1
  • ARXIV : 1305.3534


Nicolas Curien, Bénédicte Haas, Igor Kortchemski. The CRT is the scaling limit of random dissections. Random Structures and Algorithms, Wiley, 2014, pp.Online first. ⟨hal-00823219⟩



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