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

AbstracL We have derived long series expansions of the percolation probability for site and bond percolarim on d i ~ c t e d square and honeycomb lattices. For the square bond problem we have extended the sedes from 41 t e m 10 54. for the s q n m site pmblem fwm 16 terms to 37, and for the honeycomb bond problem from 13 tem to 36. Analysis of the series clearly shows that the critical exponent,6 is the same for all lhe,problems, confirming expenadow of universality. For the aitical probability and exponent we h d in lhe square bond care, qc = 0.3552994 * O.OOOWl0, p = 027643 iO.OW10. in the square site case qc = 0.294515 f O.WOOO5.,6 = 0.2763 f 0.0003; and in the hooeycomb bond case qc = 0.177 143 f O.OW002,,6 = 0.2763 * O.OW2. In addition we have obtained accurate estimates for the critical amplitudes. In all cases we find that the leadiiog correction to scang term is analytic, i.e. the confluent exponent A = 1. 1.