Coloring the Voronoi tessellation of lattices

Poisson-Voronoi percolation in the hyperbolic plane with small intensities

We consider percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. We show that the critical probability for the existence of an infinite cluster is asymptotically equal to $πλ/3$ as $λ\to0.$ This answers a question of Benjamini and Schramm.

The critical probability for Voronoi percolation in the hyperbolic plane tends to $1/2$

We consider percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. We show that the critical probability for the existence of an infinite cluster tends to $1/2$ as the intensity of the Poisson process tends to infinity. This confirms a conjecture of Benjamini and Schramm.

Tessellations of random maps of arbitrary genus

We investigate Voronoi-like tessellations of bipartite quadrangulations on surfaces of arbitrary genus, by using a natural generalization of a bijection of Marcus and Schaeffer allowing to encode such structures into labeled maps with a fixed number of faces. We investigate the scaling limits of the latter. Applications include asymptotic enumeration results for quadrangulations, and typical metric properties of randomly sampled quadrangulations. In particular, we show that scaling limits of these random quadrangulations are such that almost every pair of points are linked by a unique geodesic…

关于随机地图的镶嵌与 $t_g$-递推

On tessellations of random maps and the $t_g$-recurrence

本文研究亏格布朗曲面上 Voronoï 镶嵌胞腔质量的二阶矩，将其与 Painlevé-I 方程及 $t_g$-递推关联，并猜想其服从 Dirichlet 分布。