The Universal Amplitude Ratio $\Gamma^{-}/\Gamma^{+}$ for Two-Dimensional Percolation

TL;DR

This paper accurately calculates the universal amplitude ratio for two-dimensional percolation using novel series methods and Monte Carlo simulations, significantly narrowing down the previous wide range of estimates.

Contribution

Introduces two new series methods and combines them with Monte Carlo simulations to precisely determine the amplitude ratio in 2D percolation.

Findings

Amplitude ratio $oxed{rac{ ext{Gamma}^-}{ ext{Gamma}^+} = 162.5 \\pm 2}$

Consistent results across methods

Improved accuracy over previous estimates

Abstract

The amplitude ratio of the susceptibility (or second size-moment) for two-dimensional percolation is calculated by two series methods and also by Monte-Carlo simulation. The first series method is a new approach based upon integrating approximations to the scaling function. The second series method directly uses low- and high-density series expansions of the susceptibility, going to unprecedented orders for both bond and site percolation on the square lattice. Putting all methods together we find a consistent value $\Gamma^{-}/\Gamma^{+} = 162.5 \pm 2$, a significant improvement over previous results that placed the value of this ratio variously in the range of 14 to 220.