Low-density series expansions for directed percolation: I. A new efficient algorithm with applications to the square lattice

Abstract

A new algorithm for the derivation of low-density series for percolation on\ndirected lattices is introduced and applied to the square lattice bond and site\nproblems. Numerical evidence shows that the computational complexity grows\nexponentially, but with a growth factor $\\lambda < \\protect{\\sqrt[8]{2}}$,\nwhich is much smaller than the growth factor $\\lambda = \\protect{\\sqrt[4]{2}}$\nof the previous best algorithm. For bond (site) percolation on the directed\nsquare lattice the series has been extended to order 171 (158). Analysis of the\nseries yields sharper estimates of the critical points and exponents.\n