Chiral Potts models, friendly walkers and directed percolation problem

The X-state chiral Potts model on a finite directed lattice is defined: whose partition function under a certain boundary condition becomes the directed percolation (DP) probability for a finite lattice in the lambda --> 1 limit. We also introduce the system of m friendly walkers of L time-steps and prove that its generating function of trajectories is equal to the partition function of the X-state chiral Potts model when m = (lambda-1)/2. Combining these results gives a new formula for the DP probability given by a double limit L --> infinity and m --> O of the generating function of the m friendly walkers. We define the critical value p(c)((m)) for the infinite system of. m friendly walkers. Numerical study supports our conjecture that the critical value p(c), for the DP probability is given by the m --> O extrapolation of p(c)((m)).