Simulations of Percolation

Last modified: June 11, 2008

Here is an applet for simulating percolation on the hexagonal lattice. This is a work in progress, but seems to perform reasonably well. It will continue to be updated as improvements are made.

You can choose the dimensions of the grid between a minimum of 5-by-5 to a maximum of 100-by-100.

You can also select the two colours and p, the probability that a given hexagon is "Colour 1."

The other allowable options concern the boundary conditions:

• Fixed bottom row: colours all hexagons on the left half of the bottom row by "Colour 1" and all hexagons on the right half of the bottom row by "Colour 2"
• Fixed boundary: colours all hexagons on the left half of the rectangle by "Colour 1" and all hexagons on the right half of the rectangle by "Colour 2"
• Two boundaries: partitions the boundary into 4 sets, colouring the left and right halves of the rectangle by "Colour 1" and the middle of the top and bottom rows by "Colour 2"

Note that the boundary conditions imply that there will exist an "interface" separating "Colour 1" and "Colour 2." In the case of a fixed bottom row, the interface will start in the middle of the bottom row and end at some random spot. In the case of a fixed boundary, the interface will always connect the middle of the bottom row with the middle of the top row.

In the case of two boundaries, there will be two interfaces, although there are 2 distinct configurations that could appear. For instance, compare "25-by-25, p=0.5, seed=0, two boundaries, draw interface, partition" with "25-by-25, p=0.5, seed=3, two boundaries, draw interface, partition." In the first case, there is one cluster of "Colour 1" separating two disjoint clusters of "Colour 2," and in the second case there is one cluster of "Colour 2" separating two disjoint clusters of "Colour 1."

Checking the "Draw Interface" button will draw these interfaces and checking the "Partition" button will re-colour the grid to further illustrate how the interface separates "Colour 1" from "Colour 2."

This applet was written by Kevin Petrychyn of the University of Regina.