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

TL;DR

This paper introduces an efficient algorithm for deriving low-density series expansions in directed percolation on square lattices, significantly improving computational complexity and enabling more precise estimation of critical points and exponents.

Contribution

A novel algorithm reduces the growth factor of computational complexity for series expansions in directed percolation, extending series to higher orders than previous methods.

Findings

Growth factor of the new algorithm is less than the eighth root of 2.

Series extended to order 171 for bond and 158 for site percolation.

Sharper estimates of critical points and exponents obtained.

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 $\lambda < \protect{\sqrt[8]{2}}$, which is much smaller than the growth factor $\lambda = \protect{\sqrt[4]{2}}$ 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.