Extension of the Arrowsmith-Essam formula to the Domany-Kinzel model

Arrowsmith and Essam gave an expansion formula for point-to-point connected ness functions of the mixed site-bond percolation model on oriented lattice s, in which each term is characterized by a graph. We extend this formula r o general k-point correlation functions, which are point-to-set (with ii po ints) connectivities in the context of percolation, of the two-neighbor dis crete-time Markov process (stochastic cellular automata with two parameters ) in one dimension called the Domany-Kinzel model. which includes the mixed site-bond oriented percolation model on a square lattice as a special case . Our proof of the formula is elementary and based on induction with respec t to time-step. which is different from the original graph-theoretical one given by Arrowsmith and Essam. we introduce a system of m interacting rando m walkers called m friendly walkers (m FW) with two parameters. Following t he argument of Cardy and Colaiori, it is shown that our formula is useful t o derive a theorem that the correlation functions of the Domany-Kinzel mode l are obtained as an m --> 0 limit of the generating functions of the m FW.