Random recursive triangulations of the disk via fragmentation theory

Details Random recursive triangulations of the disk via fragmentation theory Random recursive triangulations of the disk via fragmentation theory

2010-06-04

arxiv.org/abs/1006.0792v2

Annals of Probability 2011, Vol. 39, No. 6, 2224-2270

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/10-AOP608 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

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 $\beta^*+1$, where $\beta^*=(\sqrt{17}-3)/2$, and that it can be described as the geodesic lamination coded by a random continuous function which is H\"{o}lder continuous with exponent $\beta^*-\varepsilon$, for every $\varepsilon>0$. 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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