System Size ; L =
Pc = 0.5927… ; P= 1 ×Pc
When placing black stones randomly on a square lattice, they initially appear scattered, but gradually connect to form large clusters. The way these clusters grow resembles liquid permeating through a medium. Such a problem is called as "Percolation problem", which has been actively studied for a long time.
A remarkable feature of the percolation problem is that when the occupation probability of each site reaches a critical value Pc, the size of the largest cluster diverges to infinity and spans the entire system. For square lattices, this critical value is approximately 0.5927….
In the initial state of this simulator, random numbers between [0, 1) are assigned to each grid point of an L × L square lattice. Grid points with values less than or equal to P = Pc = 0.5927… are treated as occupied sites and displayed in red.
Press the "Toggle Display" button to switch between showing occupied sites and showing clusters.
In Cluster Display mode, occupied sites that share edges are grouped into clusters and displayed in different colors. The largest cluster is shown in yellow, and other clusters are color-coded randomly in descending order of size.
The right panel shows a log-log graph of cluster sizes versus rank. This is also called a Rank-Size Plot and shows the cumulative distribution of cluster sizes. If the plot appears linear on a log-log graph, it indicates that the cluster size distribution follows a power law.
You can use the slider to change the occupation probability P from 0.95×Pc to 1.05×Pc. The random numbers assigned to each grid point remain fixed, so as P increases, previously unoccupied sites randomly become occupied. As P increases, you can see a sudden emergence of a large yellow cluster spanning the entire system around P=Pc.
Looking at the right-hand graph while changing P, you'll notice that the plot becomes linear around P=Pc, indicating that the cluster size distribution follows a power law.
Use the radio buttons to change the system size L.
Clicking the "Reset Random Numbers" button reassigns random numbers to each grid point, changing the configuration of occupied sites.