Random recursive triangulations of the disk via fragmentation theory
type: Item
instance of: scholarly article
title: Random recursive triangulations of the disk via fragmentation theory
zbMATH Open document ID: 1252.60016
DOI: 10.1214/10-AOP608
author: Nicolas Curien
author: J.-F. Le Gall
published in: The Annals of Probability
publication date: 2012-01-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/1006.0792
review text: This paper treats an infinite random triangulation of the unit disk that arises as the limit of recursive models. This triangulation is generated by throwing chords randomly in the unit disk and keeping only those chords that do not intersect with previous ones. Throwing infinitely many chords and taking the closure of the resulting set leads to a random compact set whose complement is a countable union of triangles, it is shown that this limiting set has Hausdorff dimension \(\beta^* +1\) \(=\) \((\sqrt{17} -1)/2\) and also a description as the geodesic lamination coded by a random continuous function. Lastly recursive constructions of triangulations of the \(n\)-gon that give rise to the same limit are discussed as well. More precisely, one of the peculiar features in this work consists in the use of fragmentation theory (cf. [\textit{J. Bertoin}, Random fragmentation and coagulation processes. Cambridge: Cambridge Univ. Press (2006; Zbl 1107.60002)]) to study an infinite random triangulation of the unit disk \({\mathbb D}\) that arises as the limit of several recursive models. To make the story short, let us consider a special case for explanation. Take a sequence \(U_1, V_1, U_2, V_2, \dots\) of independent random variables, which are uniformly distributed over the unit circle \({\mathbb S}_1\). \(L_1\) just consists of the chord with endpoints \(U_1\) and \(V_1\), denoted by \([U_1 V_1]\). Then at step \(n+1\), two cases are possible: either the chord \([U_{n+1} V_{n+1}]\) intersects \(L_n\), and set \(L_{n+1}\) \(=\) \(L_n\); or \([ U_{n+1} V_{n+1}]\) does not intersect \(L_n\), and set \(L_{n+1}\) \(=\) \(L_n\) \(\cup\) \([U_{n+1} V_{n+1}]\). Thus, for every integer \(n \geq 1\), \(L_n\) is a disjoint union of random chords. Let \[ L_{\infty}= \overline{ \bigcup_{n=1}^{\infty} L_n } \] be the closure of the increasing union of the sets \(L_n\). The closed set \(L_{\infty}\) is a geodesic lamination (cf. [\textit{F. Bonahon}, Contemp. Math. 269, 1--37 (2001; Z…
reviewed by: Isamu Dôku
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 05C80
zbMATH DE Number: 5995810
zbMATH Keywords: triangulation of the disk
zbMATH Keywords: noncrossing chords
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: geodesic lamination
zbMATH Keywords: fragmentation process
zbMATH Keywords: random recursive construction
MaRDI profile type: Publication
arXiv ID: 1006.0792
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description: scientific article