On the approach of the stationary state in Kauffman's random Boolean network

Kauffman's model is a Boolean network having k quenched random connections per node, and evolving deterministically in time according to a quenched random rule. For a systems of N nodes we consider a time domain 0≤t ≤ tN, with $$ tN = ∞. It is shown (i) that in this limit the average overlap between two arbitrary initial states rigorously obeys an equation first derived by Derrida and Pomeau, and (ii) that for k ≤ 2 each state evolves, in a weak sense, to a fixed point state. Also, (iii) upper bounds on the finite size corrections are obtained. Lastly, (iv) for k =1 the exact time-dependent pair overlap is found and compared to the results (i)-(iii).