Annals of Probability

Random recursive triangulations of the disk via fragmentation theory

Nicolas Curien and Jean-François Le Gall

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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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Ann. Probab., Volume 39, Number 6 (2011), 2224-2270.

First available in Project Euclid: 17 November 2011

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Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 05C80: Random graphs [See also 60B20]

Triangulation of the disk noncrossing chords Hausdorff dimension geodesic lamination fragmentation process random recursive construction


Curien, Nicolas; Le Gall, Jean-François. Random recursive triangulations of the disk via fragmentation theory. Ann. Probab. 39 (2011), no. 6, 2224--2270. doi:10.1214/10-AOP608.

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