Cluster size distribution for Bernoulli site percolation on a Poisson disc process
Channel: Nils Berglund
Date: 26/08/2022

Like the recent video https://youtu.be/zvKh0rxQgAs , this simulation shows percolation on a Poisson disc process, but this time all clusters are shown in colors depending on their size. The Poisson disc process is similar to a Poisson point process (points thrown independently and uniformly at random), but with a minimal distance between vertices. In this model, vertices are connected whenever their distance is smaller than a fixed value, chosen in such a way that the average number of neighbors is of order 5. 
Once the lattice is drawn, each vertex is given a random number between 0 and 1, following a uniform distribution, and independent of all other vertices. The parameter p varies from 0 to 1. Vertices whose value is smaller than p are considered open, the others are closed. Closed vertices are shown in dark violet, and clusters of open vertices appear in colors ranging from blue to red. The graph shows a histogram of cluster sizes. 
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. Most of them have the remarkable property that when the lattice is large, the probability of water being able to flow through the lattice changes very quickly from being very small to being close to 1. The transition occurs at a critical value pc of the parameter p, which depends on the particular lattice.
The simulation has six parts, featuring lattices of increasing size. 
110 sites: 0:00
430 sites: 0:28
1693 sites: 0:57 
6676 sites: 1:26
26529 sites: 1:55
105810 sites: 2:24
I don't actually know if the value of pc is known for Bernoulli percolation on a Poisson disc process. A reasonable guess seemed to be that pc = 1/2, but this guess may be wrong, although the simulation suggests that it is not far off. 
There exist similar but different models that have been studied, called continuum percolation models, the best-known examples being the disc model and the Boolean model, see https://en.wikipedia.org/wiki/Continuum_percolation_theory 
Note that there are two layers of randomness in the model: first the choice of the lattice, and then the choice of open/closed vertices. One cannot expect to obtain a value of pc valid for all realizations of the lattice, since there may always be very atypical configurations, that appear with a small probability. One can however try to determine the value of pc averaged over all random realizations of the lattice (this is called an annealed result), or try to show that as the lattice size goes to infinity, the value of pc will have a particular value with probability going to 1 (this is called the quenched picture).  

Render time: 30 minutes 1 second
Color scheme: Twilight by Bastian Bechtold
https://github.com/bastibe/twilight

Music: "Find Your Way Beat" by Nana Kwabena@nanakwabenamusic

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)