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

TL;DR

This paper investigates random non-crossing configurations of convex polygon diagonals, demonstrating their convergence to Brownian triangulation and refining degree distribution results using Galton-Watson tree models.

Contribution

It introduces a novel approach using conditioned Galton-Watson trees to analyze the asymptotic behavior of non-crossing configurations.

Findings

Convergence of uniform dissections and non-crossing trees to Brownian triangulation.

Refined analysis of maximal vertex degree in dissections.

Validation of a conjecture on degree distribution.

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.