| Time = 0 | |
| 10 | |
Each grid point is assigned a random number between 0 and 1, representing the resistance to fluid invasion.
For example, consider fluid invading from the left edge. Initially, the fluid invades the weakest site (smallest random number) in the leftmost column.
Next, it invades the weakest site among uninvaded boundary sites adjacent to invaded sites or still on the left edge.
In this way, the fluid invades the weakest uninvaded boundary site next to already invaded sites.
When the invaded region reaches the rightmost edge, a path is formed from left to right, and the fluid percolates through the system.
One interesting feature of invasion percolation is that the upper bound of random numbers of invaded sites converges to the percolation threshold.
Because the weakest site is always chosen, initially small random values are selected. However, as those sites become invaded, their neighbors become boundary sites, occasionally introducing smaller values again.
The sequence of invaded random values fluctuates, but the upper bound converges to the percolation threshold.
This simulator plots random numbers of invaded sites. The plot shows that the threshold line (0.5927...) closely matches the upper limit of selected values.
Reference:
D Wilkinson and JF Wilemsen, J. Phys. A: Math. Gen. 16 (1983) 3365
"Invasion Percolation: a new form of percolation theory"