The dual tree of a recursive triangulation of the disk

Broutin, Nicolas ^[1] ; Sulzbach, Henning. ^[2]
1. [1] French Institute for Research in Computer Science and Automation
  
  French Institute for Research in Computer Science and Automation
  
  Arrondissement de Versailles, Francia
2. [2] McGill University
  
  McGill University
  
  Canadá
Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 43, Nº. 2, 2015, págs. 738-781
Idioma: inglés
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Resumen
- 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 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 T, which is encoded by M. This confirms a conjecture of Curien and Le Gall.