Distribution of poles in a series expansion of the asymmetric directed-bond percolation probability on the square lattice

We investigate numerically the percolation probability of the asymmetric directed-bond percolation on the square lattice with two parameters p and q based on Guttmann and Enting's procedure (1996 Phys. Rev. Lett. 76 344). A series in the form of Sigma(n) H-n(q) p(n) is derived by using the finite transfer-matrix method. The denominator of H-n(q) is directly calculated from the determinant of the transfer matrix and it leads to a proof that poles ail lies on the unit circle in the complex q plane. The solvability of the bond directed percolation is also discussed.