The Annals of Probability

The dual tree of a recursive triangulation of the disk

Nicolas Broutin and Henning Sulzbach

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Abstract

In the recursive lamination of the disk, one tries to add chords one after another at random; a chord is kept and inserted if it does not intersect any of the previously inserted ones. Curien and Le Gall [Ann. Probab. 39 (2011) 2224–2270] have proved that the set of chords converges to a limit triangulation of the disk encoded by a continuous process $\mathscr{M}$. Based on a new approach resembling ideas from the so-called contraction method in function spaces, we prove that, when properly rescaled, the planar dual of the discrete lamination converges almost surely in the Gromov–Hausdorff sense to a limit real tree $\mathscr{T}$, which is encoded by $\mathscr{M}$. This confirms a conjecture of Curien and Le Gall.

Article information

Source
Ann. Probab., Volume 43, Number 2 (2015), 738-781.

Dates
First available in Project Euclid: 2 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aop/1422885574

Digital Object Identifier
doi:10.1214/13-AOP894

Mathematical Reviews number (MathSciNet)
MR3306004

Zentralblatt MATH identifier
1355.60014

Subjects
Primary: 60C05: Combinatorial probability 60F17: Functional limit theorems; invariance principles 05C05: Trees
Secondary: 11Y16: Algorithms; complexity [See also 68Q25] 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 05A16: Asymptotic enumeration

Keywords
Real tree Gromov–Hausdorff convergence functional limit theorem contraction method

Citation

Broutin, Nicolas; Sulzbach, Henning. The dual tree of a recursive triangulation of the disk. Ann. Probab. 43 (2015), no. 2, 738--781. doi:10.1214/13-AOP894. https://projecteuclid.org/euclid.aop/1422885574


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