Growth of one cluster in Bernoulli percolation on a cubic lattice
Channel: Nils Berglund
Date: 19/09/2022

This second simulation of percolation on a cubic lattice shows the growth of one percolation cluster, the one adjacent to one face of the large cube. In less technical terms, if percolation is used to model water flowing through porous rock, the video shows the volume occupied by water, when the rock has been made transparent. The color gradient shows how deep the water has penetrated into the rock. 
There are seven parts in this video, showing lattices of increasing size:
4x4x4 lattice: 0:00
8x8x8 lattice: 0:31
16x16x16 lattice: 1:02
32x32x32 lattice: 1:33
64x64x64 lattice: 2:04
128x128x128 lattice: 2:35
256x256x256 lattice: 3:07
The cubes forming the left back side of the large cube are all open. All other small cubes are open with probability p, and closed with probability 1-p, where the parameter p increases from 0 to 1 in the course of the each part of the video. All cubes are open or closed independently of all others.  
Only flooded cubes are shown. These are the cubes that can be reached from the left back face of the large cube by moving only through open cubes. 
The graph shows the ratio between the number of flooded cubes and the number of open cubes.
Percolation models have the particularity that there is a critical value pc of the probability p, such that, when p is smaller than pc, there are many small clusters, and it is unlikely that the cluster of flooded cells reaches the face opposite to the initially flooded face. When p is larger than pc, on the other hand, if is very likely that there is one giant cluster of flooded cells, meaning that water is able to flow through the system. Numerical simulations show that the critical probability pc is close to 0.311604, although its exact value is not known. The transition between the two regimes becomes sharper as the number of small cubes increases. 

Render time: 53 minutes 38 seconds
Color scheme: Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
https://github.com/BIDS/colormap

Music: Cycles by Audionautix is licensed under a Creative Commons Attribution 4.0 licence. https://creativecommons.org/licenses/by/4.0/
Artist: http://audionautix.com/

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)