Open Access
November 2011 Random recursive triangulations of the disk via fragmentation theory
Nicolas Curien, Jean-François Le Gall
Ann. Probab. 39(6): 2224-2270 (November 2011). DOI: 10.1214/10-AOP608


We introduce and study an infinite random triangulation of the unit disk that arises as the limit of several recursive models. This triangulation is generated by throwing chords uniformly at random in the unit disk and keeping only those chords that do not intersect the previous ones. After throwing infinitely many chords and taking the closure of the resulting set, one gets a random compact subset of the unit disk whose complement is a countable union of triangles. We show that this limiting random set has Hausdorff dimension β* + 1, where β* = (√17 − 3)/2, and that it can be described as the geodesic lamination coded by a random continuous function which is Hölder continuous with exponent β* − ε, for every ε > 0. We also discuss recursive constructions of triangulations of the n-gon that give rise to the same continuous limit when n tends to infinity.


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Nicolas Curien. Jean-François Le Gall. "Random recursive triangulations of the disk via fragmentation theory." Ann. Probab. 39 (6) 2224 - 2270, November 2011.


Published: November 2011
First available in Project Euclid: 17 November 2011

zbMATH: 1252.60016
MathSciNet: MR2932668
Digital Object Identifier: 10.1214/10-AOP608

Primary: 05C80 , 60D05 , 60J80

Keywords: Fragmentation process , geodesic lamination , Hausdorff dimension , noncrossing chords , random recursive construction , Triangulation of the disk

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.39 • No. 6 • November 2011
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