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 directed lattices is introduced and applied to the square lattice bond and site problems. Numerical evidence shows that the computational complexity grows exponentially, but with a growth factor $λ< \protect{\sqrt[8]{2}}$, which is much smaller than the growth factor $λ= \protect{\sqrt[4]{2}}$ of the previous best algorithm. For bond (site) percolation on the directed square lattice the series has been extended to order 171 (158). Analysis of the series yields sharper estimates of the critical points and exponents.