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., vol. 39, pp. 2224‐2270, 2011] 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. 1 Introduction and main results In [18], Curien and Le Gall introduce the model of random recursive triangulations of the disk. The construction goes as follows: Atn = 1, two points are sampled independently with uniform distribution on the circle. They are connected by a chord (a straight line) which splits the disk into two fragments. Later on, at each step, two independent points are sampled uniformly at random on the circle and are connected by a chord if the latter does not intersect any of the previously inserted chords; in other words the two points are connected by a chord if they both fall in the same fragment. This gives rise to a sequence of laminations of the disk; for us a lamination will be a collection of chords which may only intersect at their end points. At timen the lamination Ln consists of the union of the chords inserted up to timen. As an increasing closed subset of the disk, Ln converges, and it is proved in [18] that
@cite_27: We study various models of random non-crossing configurations consisting of diagonals of convex polygons, and focus in particular on uniform dissections and non-crossing trees. For both these models, we prove convergence in distribution towards Aldous' Brownian triangulation of the disk. In the case of dissections, we also refine the study of the maximal vertex degree and validate a conjecture of Bernasconi, Panagiotou and Steger. Our main tool is the use of an underlying Galton-Watson tree structure. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 236-260, 2014
@cite_8: A noncrossing tree (NC-tree) is a tree drawn on the plane having as vertices a set of points on the boundary of a circle, and whose edges are straight line segments that do not cross. In this article, we show that NC-trees with size n are conditioned Galton-Watson trees. As corollaries, we give the limit law of depth-first traversal processes and the limit profile of NC-trees.
Among the recent work on laminations of the disk, one can mention @cite_27 where Curien and Kortchemski showed that the Brownian triangulation is also the scaling limit of other random subsets of the disk, in particular non-crossing trees (sets of non-crossing chords which form a tree) @cite_8 , and dissections (non-crossing sets of chords) under the uniform distribution. By sampling tessellations according to a Boltzmann weight depending on the degree of the faces, obtained limit laminations which are not triangulations and are encoded by excursions of stable spectrally positive L 'evy processes (with L 'evy measure concentrated on @math ). Finally, have studied geodesic laminations of the Poincar 'e disk. They construct and study the unique random tiling of the hyperbolic plane into triangles with vertices on the boundary whose distribution is invariant under M "obius transformations and satisfies a certain spatial Markov property.