Random non‐crossing plane configurations: A conditioned Galton‐Watson tree approach

Abstract

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.

Random stable laminations of the disk

“…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 of uniform dissections of P n have also been studied, extending the previously mentioned results for triangulations (see [3,10]). From a more geometrical point of view, Aldous [1,2] studied the shape of a large uniform triangulation viewed as a random compact subset of the closed unit disk.…”

“…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).…”

“…degenerate in the limit. See also the recent paper [10] for other classes of noncrossing configurations of the polygon that converge to the Brownian triangulation.…”

“…(ii) In [10], it is shown that Theorem 3.1 can be used to study uniformly distributed dissections. More precisely, if one sets µ 0 = 2 − √ 2 and µ i = ((2 − √ 2)/2) i−1 for every i ≥ 2, then the Boltzmann probability measure P µ n associated to µ is the uniform probability measure on dissections of P n+1 .…”

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.…”