Tessellations of random maps of arbitrary genus

TL;DR

This paper studies Voronoi-like tessellations of bipartite quadrangulations on surfaces of any genus, exploring their encoding, scaling limits, and metric properties, with applications in enumeration and understanding geodesic structures.

Contribution

It introduces a generalized bijection for encoding quadrangulations on arbitrary genus surfaces and analyzes their scaling limits and metric properties.

Findings

Scaling limits show unique geodesic connections between almost all point pairs.

Provides asymptotic enumeration results for quadrangulations.

Demonstrates typical metric properties of random quadrangulations.

Abstract

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.