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Article Dans Une Revue Random Structures and Algorithms Année : 2015

The CRT is the scaling limit of random dissections

Résumé

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.

Dates et versions

hal-00823219 , version 1 (16-05-2013)

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Nicolas Curien, Bénédicte Haas, Igor Kortchemski. The CRT is the scaling limit of random dissections. Random Structures and Algorithms, 2015, 47 (2), pp.304-327. ⟨10.1002/rsa.20554⟩. ⟨hal-00823219⟩
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