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The CRT is the scaling limit of random dissections

Curien, Nicolas; Haas, Bénédicte; Kortchemski, Igor (2015), The CRT is the scaling limit of random dissections, Random Structures & Algorithms, 47, 2, p. 304-327. 10.1002/rsa.20554

Type
Article accepté pour publication ou publié
External document link
https://arxiv.org/abs/1305.3534v2
Date
2015
Journal name
Random Structures & Algorithms
Volume
47
Number
2
Publisher
J. Wiley
Pages
304-327
Publication identifier
Metadata
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Author(s)
Curien, Nicolas

Haas, Bénédicte
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Kortchemski, Igor
Abstract (EN)
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.
Subjects / Keywords
Brownian Continuum Random Tree; Gromov–Hausdorff topology; Random dissections; Galton–Watson trees; scaling limits