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_31: Novel substituted 2-pyrones IN WHICH ONE OF THE RADICALS R1 to R4 is substituted by an acryloyl or methacryloyl group are obtained from the corresponding 2-pyrone derivatives and acrylic or methacrylic acid or a derivative thereof. The compounds can be polymerized forming light-sensitive polymers which are cross-linked by the action of light and may be used in light-sensitive copying layers.
@cite_23: The hydrochloride and hydrobromide of aspartic anhydride are obtained in high yields when the known reaction between one mole of aspartic acid and suitable halide dehydrating agents, such as phosphorus chlorides, is carried out in the presence of at least one mole of a lower alkanoic acid which provides at least a portion of the liquid reaction medium. The yield is further improved by adding the anhydride of the alkanoic acid to the reaction mixture.
@cite_7: We construct and study the unique random tiling of the hyperbolic plane into ideal hyperbolic triangles (with the three corners located on the boundary) that is invariant (in law) with respect to Mobius transformations, and possesses a natural spatial Markov property that can be roughly described as the conditional independence of the two parts of the triangulation on the two sides of the edge of one of its triangles.
The work of was motivated by the pioneering work of who studied of the disk which arise as limiting objects for uniform triangulations of regular @math -gons as @math . In the case of uniform random triangulations, the process which encodes the limit triangulation is the Brownian excursion, and the scaling limit of the sequence of dual trees is the Brownian continuum random tree introduced in @cite_7 @cite_23 @cite_31 .