Series expansions of the percolation probability for directed square and honeycomb lattices

Abstract

We have derived long series expansions of the percolation probability for site and bond percolation on directed square and honeycomb lattices. For the square bond problem we have extended the series from 41 terms to 54, for the square site problem from 16 terms to 37, and for the honeycomb bond problem from 13 terms to 36. Analysis of the series clearly shows that the critical exponent $\beta$ is the same for all the problems confirming expectations of universality. For the critical probability and exponent we find in the square bond case, $q_c = 0.3552994\pm 0.0000010$, $\beta = 0.27643\pm 0.00010$, in the square site case $q_c = 0.294515 \pm 0.000005$, $\beta = 0.2763 \pm 0.0003$, and in the honeycomb bond case $q_c = 0.177143 \pm 0.000002$, $\beta = 0.2763 \pm 0.0002$. In addition we have obtained accurate estimates for the critical amplitudes. In all cases we find that the leading correction to scaling term is analytic, i.e., the confluent exponent $\Delta = 1$.