Local properties of the random Delaunay triangulation model and topological 2D gravity

TL;DR

This paper explores the geometric and topological properties of random Delaunay triangulations, linking them to moduli space measures and demonstrating volume growth properties that may connect to Liouville conformal field theory.

Contribution

It establishes a connection between Delaunay triangulation measures and the Weil-Petersson form, and proves volume monotonicity as points are added, advancing understanding of large-scale triangulation limits.

Findings

Relation between Delaunay measure and Weil-Petersson form

Maximality property of Delaunay triangulations

Volume increases monotonically with added points

Abstract

Delaunay triangulations provide a bijection between a set of $N+3$ points in the complex plane, and the set of triangulations with given circumcircle intersection angles. The uniform Lebesgue measure on these angles translates into a K\"ahler measure for Delaunay triangulations, or equivalently on the moduli space $\mathcal M_{0,N+3}$ of genus zero Riemann surfaces with $N+3$ marked points. We study the properties of this measure. First we relate it to the topological Weil-Petersson symplectic form on the moduli space $\mathcal M_{0,N+3}$. Then we show that this measure, properly extended to the space of all triangulations on the plane, has maximality properties for Delaunay triangulations. Finally we show, using new local inequalities on the measures, that the volume $\mathcal{V}_N$ on triangulations with $N+3$ points is monotonically increasing when a point is added, $N\to N+1$. We expect that this can be a step towards seeing that the large $N$ limit of random triangulations can tend to the Liouville conformal field theory.