Series expansions of the percolation probability on the directed triangular lattice

TL;DR

This paper derives extensive series expansions for percolation probabilities on the directed triangular lattice, providing precise estimates of critical points and exponents, and confirming universality across different percolation types.

Contribution

The authors extended series expansions for site, bond, and site-bond percolation on the directed triangular lattice, including the first series for the site-bond case, and provided accurate critical parameters.

Findings

Critical exponents agree with square lattice results.

Precise critical probabilities for each percolation type.

Leading correction to scaling is analytic with exponent Δ=1.

Abstract

We have derived long series expansions of the percolation probability for site, bond and site-bond percolation on the directed triangular lattice. For the bond problem we have extended the series from order 12 to 51 and for the site problem from order 12 to 35. For the site-bond problem, which has not been studied before, we have derived the series to order 32. Our estimates of the critical exponent $\beta$ are in full agreement with results for similar problems on the square lattice, confirming expectations of universality. For the critical probability and exponent we find in the site case: $q_c = 0.4043528 \pm 0.0000010$ and $\beta = 0.27645 \pm 0.00010$; in the bond case: $q_c = 0.52198\pm 0.00001$ and $\beta = 0.2769\pm 0.0010$; and in the site-bond case: $q_c = 0.264173 \pm 0.000003$ and $\beta = 0.2766 \pm 0.0003$. 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 $\Delta = 1$.