Universal critical amplitude ratios for percolation

The hypothesis of universality implies that for every scaling relation among critical exponents there exists a universal ratio among the corresponding critical amplitudes. If one writes $B{\mathrm{}\left|t\right|\mathrm{}}^{\beta }$, ${A_F}^{\pm }{\mathrm{}\left|t\right|\mathrm{}}^{2-\alpha }$, $C^{\pm }{\mathrm{}\left|t\right|\mathrm{}}^{-\gamma }$, and ${\xi }_0{\mathrm{}\left|t\right|\mathrm{}}^{-\nu }$ [where $t=\frac{(p_c-p)}{p_c}$, $p$ being the concentration of nonzero bonds, and +(-) stands for $p<\mathrm{}{p}_c$ ($p>\mathrm{}{p}_c$)] for the leading singular terms in the probability to belong to the infinite cluster, the mean number of clusters, the clusters' mean-square size, and the pair connectedness correlation length, then it is shown that the ratios $\frac{{A_F}^{+}}{{A_F}^{-}}$, $\frac{C^{+}}{C^{-}}$, ${A_F}^{+}{B}^{-2\mathrm{}}{C}^{+}$, $\frac{{\xi }_0^{+}}{{\xi }_0^{-}}$, and ${A_F}^{+}{\left({\xi }_0^{+}\right)}^d$ ($d$ is the dimensionality) are universal. Similar quantities are found for the behavior at $p=p_c$ (as a function of a "ghost" field). All of these universal ratios are derived from a universal scaled equation of state, which is calculated to second order in $\epsilon =6-d$. The (extrapolated) results are compared with available information in dimensionalities $d=2, 3, 4, 5$, with reasonable agreements. The amplitude relations become exact at $d=6\mathrm{}$, when logarithmic corrections appear. Additional universal ratios are obtained for the confluent correction to scaling terms.