# Homogenization of random quasiconformal mappings and random Delauney triangulations

## TL;DR

This paper investigates the homogenization of random media, demonstrating that random quasiconformal mappings approximate affine maps and random Delaunay triangulations approximate conformal maps, confirming existing conjectures.

## Contribution

It provides new results on the approximation of random quasiconformal mappings and Delaunay triangulations by simpler geometric structures, confirming a conjecture of Stephenson.

## Key findings

- Random quasiconformal mappings are close to affine mappings.
- Circle packings of random Delaunay triangulations approximate conformal maps.
- On Riemann surfaces, random Delaunay triangulations are nearly circle packed.

## Abstract

In this paper, we solve two problems dealing with the homogenization of random media. We show that a random quasiconformal mapping is close to an affine mapping, while a circle packing of a random Delauney triangulation is close to a conformal map, confirming a conjecture of Stephenson. We also show that on a Riemann surface equipped with a conformal metric, a random Delauney triangulation is close to being circle packed.