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March 2015 The dual tree of a recursive triangulation of the disk
Nicolas Broutin, Henning Sulzbach
Ann. Probab. 43(2): 738-781 (March 2015). DOI: 10.1214/13-AOP894

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.

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Nicolas Broutin. Henning Sulzbach. "The dual tree of a recursive triangulation of the disk." Ann. Probab. 43 (2) 738 - 781, March 2015. https://doi.org/10.1214/13-AOP894

Information

Published: March 2015
First available in Project Euclid: 2 February 2015

zbMATH: 1355.60014
MathSciNet: MR3306004
Digital Object Identifier: 10.1214/13-AOP894

Subjects:
Primary: 05C05, 60C05, 60F17
Secondary: 05A15, 05A16, 11Y16

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 2 • March 2015
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