# On a conjecture by Chapuy about Voronoi cells in large maps

## TL;DR

This paper confirms Chapuy's conjecture for the case of two points in large planar maps, showing the area ratio of Voronoi cells is uniformly distributed, thus supporting the conjecture's validity in these cases.

## Contribution

It provides a direct computation confirming the conjecture for k=2 in large planar maps, extending previous partial results.

## Key findings

- The area ratio of Voronoi cells is uniformly distributed for large planar maps.
- Chapuy's conjecture holds for k=2 in large planar quadrangulations and general planar maps.
- The result supports the conjecture's validity in specific cases.

## Abstract

In a recent paper, Chapuy conjectured that, for any positive integer k, the law for the fractions of total area covered by the k Voronoi cells defined by k points picked uniformly at random in the Brownian map of any fixed genus is the same law as that of a uniform k-division of the unit interval. For k=2, i.e. with two points chosen uniformly at random, it means that the law for the ratio of the area of one of the two Voronoi cells by the total area of the map is uniform between 0 and 1. Here, by a direct computation of the desired law, we show that this latter conjecture for k=2 actually holds in the case of large planar (genus 0) quadrangulations as well as for large general planar maps (i.e. maps whose faces have arbitrary degrees). This corroborates Chapuy's conjecture in its simplest realizations.