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

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.