Random stable laminations of the disk

Abstract

We study large random dissections of polygons. We consider random dissections of a regular polygon with n sides, which are chosen according to Boltzmann weights in the domain of attraction of a stable law of index θ ∈ (1, 2]. As n goes to infinity, we prove that these random dissections converge in distribution toward a random compact set, called the random stable lamination. If θ = 2, we recover Aldous' Brownian triangulation. However, if θ ∈ (1, 2), large faces remain in the limit and a different random compact set appears. We show that the random stable lamination can be coded by the continuous-time height function associated to the normalized excursion of a strictly stable spectrally positive Lévy process of index θ. Using this coding, we establish that the Hausdorff dimension of the stable random lamination is almost surely 2 − 1/θ.Introduction. In this article we study large random dissections of polygons. A dissection of a polygon is the union of the sides of the polygon and of a collection of diagonals that may intersect only at their endpoints. The faces are the connected components of the complement of the dissection in the polygon. The particular case of triangulations (when all faces are triangles) has been extensively studied in the literature. For every integer n ≥ 3, let P n be the regular polygon with n sides whose vertices are the nth roots of unity. It is well known that the number of triangulations of P n is the Catalan number of order n − 2. In the general case, where faces of degree greater than three are allowed, there is no known explicit formula for the number of dissections of P n , although an asymptotic estimate is known (see [10,17]). Probabilistic aspects of uniformly distributed random triangulations have been investigated; see, for example, the articles [18,19] which study graphtheoretical properties of uniform triangulations (such as the maximal vertex degree or the number of vertices of degree k). Graph-theoretical properties

Dissecting the circle, at random

“…Using this topology, the Brownian triangulation appears as a universal limit of uniform non-crossing configurations: Theorem 1 ( [17] and [9]). If C n is a uniform n.c.c.…”

“…See [2,9,17] for complete arguments. This is in essence the idea of the proof of Theorem 1: for each class of non-crossing configurations, we find a bijection (usually the classical duality operation) with a class of trees.…”

“…The results of this section come from [17] to which the reader is referred for details. As we saw in the last section, the universal limit of various classes of uniform non-crossing configurations is a random continuous triangulation.…”

“…The main difficulty that arises when θ < 2 is that these random excursion functions do not have distinct local minima (hypothesis H e ) and thus the construction of last section breaks down. Still, it is possible to build the stable lamination from the stable height process in a way similar as L e is constructed from e, see [17] for more details. Many distributional properties of the stable laminations are still to be calculated.…”

“…This fractal object (it almost surely has Hausdorff dimension 3/2) has a fascinating structure and is connected to the Brownian continuum random tree [1] which can be thought of as its dual. The work of Aldous opened the doors for understanding the geometric structure of a large variety of random non-crossing structures yielding to a number of new objects such as the stable laminations ( [17] and Section 1.2), the recursive triangulation ( [10] and Section 2) or the Markovian hyperbolic triangulation ( [11] and Section 4).…”