Hypergeometric series in a series expansion of the directed-bond percolation probability on the square lattice

The asymmetric directed-bond percolation (ADBP) problem with an asymmetry parameter k is introduced and some rigorous results are given concerning a series expansion of the percolation probability on the square lattice. It is shown that the first correction term d(n,l)(k) is expressed by Gauss' hypergeometric series with a variable k. Since the ADBP includes the ordinary directed bond percolation as a special case with k = 1, our results give another proof for the Baxter-Guttmann's conjecture that d(n,1)(1) is given by the Catalan number, which was recently proved by Bousquet-Melou. Direct calculations on finite lattices are performed and combining them with the present results determines the first 14 terms of the series expansion for percolation probability of the ADBP on the square lattice. The analysis by Dlog Pade approximations suggests that the critical value depends on k, while asymmetry does not change the critical exponent beta of percolation probability.