Galton-Watson tree의 구현입니다. 그림의 중심에는 최초의 공통 조상이 2개 자손을 갖고 있습니다. 각 개체의 자손 수는 (0에서 3 사이의) 상호독립적인 같은 분포를 갖습니다.

자세한 설명은 http://images. math. cnrs.fr/La-probabilite-d-extinction-d-une.html 을 참조하세요. (프랑스어)

Z2의 부분집합에서 결합 여과(bond percolation)의 모습 구현 (p = 0.525).

각각의 간선들은 확률 p로 열려있고, 어둡게 칠해진 간선들은 전체 그림의 아래쪽 경계를 포함하는 연결부 (connected component)를 나타냅니다.

입자들은 무작위로 떨어지고 다른 입자의 아래 혹은 옆에 놓이자마자 정지합니다. 이 모델의 극한은 KPZ 방정식에 의해 나타납니다.

Voir aussi http://images. math. cnrs.fr/Qu-est-ce-qu-une-Equation-aux.html

확률적 Allen-Cahn 방정식은 합금에서의 상분리(phase separation)를 설명합니다.

Voir aussi http://images. math. cnrs.fr/Qu-est-ce-qu-une-Equation-aux.html

Benoulli site percolation on a square lattice of size 992 x 1920. Each site is open with probability 0.5997, and closed otherwise, independently of all other sites. Closed sites are shown in yellow. Open sites connected to the left boundary are shown in purple, the other open sites in cyan. The critical value of Bernoulli site percolation on a square lattice is approximately 0.59274. Beyond this value, as the lattice size goes to infinity, there is an infinite open cluster with probability 1. Below this value, the probability is 0. 

Benoulli bond percolation on a square lattice of size 992 x 1920. Each site is open with probability 0.4993, and closed otherwise, independently of all other sites. Closed sites are shown in yellow. Open sites connected to the left boundary are shown in blue, the other open sites in cyan. The critical value of Bernoulli site percolation on a square lattice is equal to 1/2. Beyond this value, as the lattice size goes to infinity, there is an infinite open cluster with probability 1. Below this value, the probability is 0.

Benoulli percolation on a honeycomb lattice of size 71 x 119. Each hexagon is open with probability 0.4823, and closed otherwise, independently of all other hexagons. Closed hexagons are shown in blue. Open hexagons connected to the left boundary are shown in yellow, the other open hexagons in gray. The critical value of Bernoulli percolation on a honeycomb lattice is equal to 1/2. Beyond this value, as the lattice size goes to infinity, there is an infinite open cluster with probability 1. Below this value, the probability is 0.

Clusters of Bernoulli site percolation on a honeycomb lattice. Connected open components are shown in different colors. The probability that a bond is open is equal to 0.4978.

The cells of the lattice are equilateral triangles. Their centers form a hexagonal lattice, whose Voronoi cells are the triangles.

The parameter p  = 0.6955 is the probability that a triangle is open. Closed triangles are shown in purple, and open ones in yellow when they are connected to the left boundary, and pink otherwise. The increase of p slows down when it reaches the critical value, which is approximately equal to 0.69704. The graph shows the rsize of the percolation cluster (the set of open triangles connected to the left boundary), divided by the number of open triangles, as a function of p.

The parameter p increasing from 0 to 1 is the probability that a bond is open. Closed bonds are shown in dark purple, and open ones in yellow when they are connected to the left boundary, and pink otherwise. The increase of p slows down when it reaches the critical value, which is equal to 1 - 2*sin(Pi/18), or about 0.652703645. The graph shows the rsize of the percolation cluster (the set of open bonds connected to the left boundary), divided by the number of open bonds, as a function of p.

Clusters of Bernoulli site percolation on a honeycomb lattice. Connected open components are shown in different colors. The probability that a bond is open is equal to 0.5041.

Clusters of a percolation configuration on a lattice of triangles (whose centers form a hexagonal lattice). The probability of a triangle to be open is equal to 0.69704, which is close to the numerically determined critical value. The clusters are therefore statistically self-similar.