# A universal law for Voronoi cell volumes in infinitely large maps

## TL;DR

This paper investigates the distribution of Voronoi cell volumes in large random planar quadrangulations, revealing a universal scaling law and providing an explicit formula for the distribution's Laplace transform.

## Contribution

It introduces a universal law for the scaled volume of a Voronoi cell in infinite maps and derives an explicit Laplace transform for its distribution.

## Key findings

- The finite Voronoi cell volume scales as s^4 for large s.
- The distribution of the scaled volume is universal across models.
- An explicit formula for the Laplace transform of the volume distribution is provided.

## Abstract

We discuss the volume of Voronoi cells defined by two marked vertices picked randomly at a fixed given mutual distance 2s in random planar quadrangulations. We consider the regime where the mutual distance 2s is kept finite while the total volume of the quadrangulation tends to infinity. In this regime, exactly one of the Voronoi cells keeps a finite volume, which scales as s^4 for large s. We analyze the universal probability distribution of this, properly rescaled, finite volume and present an explicit formula for its Laplace transform.