Percolation, on the occasion of Hugo Duminil-Copin's Fields Medal
Channel: Nils Berglund
Date: 11/07/2022

I was delighted to hear that last week, my fellow mathematician Hugo Duminil-Copin (IHES Paris and University of Geneva) was awarded the Fields Medal for his work on phase transitions in statistical physics. To find out more about Hugo's research, I recommend reading Martin Hairer's short review available here: https://www.researchgate.net/publication/361788066_The_work_of_Hugo_Duminil-Copin - Edit: You can also watch a talk given by Hugo at the ICM 2018 in Rio here: https://youtu.be/kNn81dGhFak
Among the topics of Hugo's work are percolation models. This provided me with a good opportunity of making new versions of a couple of old percolation simulations on this channel, such as https://youtu.be/cI_B9iqsB9E
Percolation models describe the ability of a liquid to flow through a porous medium, such as oil flowing through certain types of rock, or water flowing through a coffee machine. The relatively simple model shown here is called Bernoulli site percolation on a square lattice. Each site of the lattice is open with probability p, and closed with probability 1-p. Closed sites are shown in yellow. Open sites can be flooded (shown in purple), or not (shown in cyan). The flooded sites are those of the left column, and those that can be reached from the left column by moving only through open sites.
The simulation has six parts, featuring lattices of increasing size. 
31x60 grid: 0:00
62x120 grid: 0:41
124x240 grid: 1:22
248x480 grid: 2:03
496x960 grid: 2:44
992x1920 grid: 3:24
In each part, the probability p of sites being open increases smoothly from 0 to 1. The remarkable property of percolation models is that there is a rather sharp transition between cases where only very few sites are flooded, to cases where most of them are flooded. The transition becomes sharper as the size of the lattice increases. One can show that when the lattice size goes to infinity, the transition concentrates on a particular value pc of p, which depends on the type of lattice. For the square lattice shown here, the critical value of p is close to 0.59274, see https://en.wikipedia.org/wiki/Percolation_threshold . To make transitions more visible, in each part the increase of p is slowed down around this critical value. Of particular interest is the shape of the cluster of flooded cells when p is equal to pc, which has fractal properties. 
There is no closed mathematical formula known for the critical probability pc in Bernoulli site percolation. However, for bond percolation, which is shown in the simulation https://youtu.be/cI_B9iqsB9E , pc has been shown to be equal to 0.5. There is an intuitive explanation for that based on "duality": one compares percolation on the initial lattice, and on the dual lattice, whose bonds have been rotated by a right angle. 
In each part of this simulation, all sites are given random values between 0 and 1. These values are independent, and follow a uniform distribution. A site becomes open when its value is smaller than the increasing parameter p. Then a variant of depth-first search algorithm is used to find all flooded cells. 

Render time: 17 minutes 20 seconds
Color scheme: Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
https://github.com/BIDS/colormap

Music: "Hang Ups (Want You)" by Otis McDonald

Current version of the C code used to make these animations: 
https://github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html

Some outreach articles on mathematics:
https://images.math.cnrs.fr/_Berglund-Nils-1343_.html
(in French, some with a Spanish translation)