Percolation thresholds in square-lattice Kauffman model.

In Kauffman's random Boolean network model for genetics, each gene is either on or off, depending in a fixed random way on whether K neighbor genes are on or off. Our computer simulation puts these genes on the sites of a square lattice and asks if the "off" genes, the "on" genes, the "oscillating" genes and the non-oscillating "stable" genes are percolating, i.e. form one connected network of neighboring sites. The percolation thresholds for stable and for oscillating genes are found to coincide numerically with the transition to chaos at p = 0.29. Up to one million sweeps through the lattice were made to find that agreement.