The Annals of Probability

Random recursive triangulations of the disk via fragmentation theory

Nicolas Curien and Jean-François Le Gall

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 17 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1321539120

Digital Object Identifier
doi:10.1214/10-AOP608

Mathematical Reviews number (MathSciNet)
MR2932668

Zentralblatt MATH identifier
1252.60016

Subjects
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]

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

Citation

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. https://projecteuclid.org/euclid.aop/1321539120


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