We have used a slightly modified version of Roskies's quadratic map method [Phys. Rev. B 24, 5305 (1981)] to estimate the confluent correction amplitudes for the susceptibility $\ensuremath{\chi}$ and correlation length $\ensuremath{\xi}$ of a model that interpolates between Ising and Gaussian limits. Our best estimates for the corresponding leading exponents are $\ensuremath{\gamma}=1.237\ifmmode\pm\else\textpm\fi{}0.003$, $\ensuremath{\nu}=0.630\ifmmode\pm\else\textpm\fi{}0.003$, and $\ensuremath{\eta}=2\ensuremath{-}\frac{\ensuremath{\gamma}}{\ensuremath{\nu}}=0.036\ifmmode\pm\else\textpm\fi{}0.002$. The correction to scaling amplitude ratio $\frac{B(\ensuremath{\xi})}{B(\ensuremath{\chi})}=0.8\ifmmode\pm\else\textpm\fi{}0.1$ is slightly larger than that predicted from ${\ensuremath{\varphi}}^{4}$ continuum model analysis [M.-c. Chang and A. Houghton, Phys. Rev. Lett. 44, 785 (1980) and C. Bagnuls and C. Bervillier, Phys. Rev. B 24, 1226 (1981)]
We have extended the series for the zero-field susceptibility of the spin-
s Ising model through tenth order in the reduced inverse temperature
K on the square, triangular, simple-cubic, body-centered-cubic, and face-centered-cubic lattices. The series coefficients
hn(s) are expressed as simple polynomials in
X=s(s+1). Using extended methods of analysis we have estimated the nature of the leading singularities on the face-centered lattice and conclude with good confidence that the susceptibility exponent ${\ensuremath{\gamma}}_{1}$ equals 1.25, independent of
s. The exponent of the leading correction term is estimated to be ${\ensuremath{\gamma}}_{2}\ensuremath{\simeq}0.75\ifmmode\pm\else\textpm\fi{}0.08$ in good agreement with renormalization-group theory. For
s=21 only, the amplitude of the confluent correction apparently vanishes. We have also studied the leading singularities on the triangular lattice and conclude that ${\ensuremath{\gamma}}_{1}=\frac{7}{4}$, independent of
s, with, however, much stronger corrections than in three dimensions. These results provide a very strong corroboration of the universality hypothesis.
A quadratic map that was recently proposed by Roskies in order to properly treat the confluent singularities in the three-dimensional Ising model, has been generalized to an unbiased map. This map (which does not require an input of the confluent exponent ${\ensuremath{\Delta}}_{1}$, from renormalization-group calculations) is employed here to obtain independent series estimates of $\ensuremath{\gamma}$, $\ensuremath{\nu}$ and ${\ensuremath{\Delta}}_{1}$ for the three-dimensional Ising model.
Discoveries in physics often hinge on a disagreement between theory and experiment of a percent or less. Because of such a discrepancy, many workers in statistical mechanics have been concerned about the complete validity of the scaling and renormalization-group theories of critical phenomena. For the three-dimensional Ising model, the experimentally derived values for the critical exponents in fluids have agreed to three significant figures with those predicted by the renormalization group. However, the predictions disagreed by a couple of percent with exponents calculated by series expansions, and the claimed accuracy of these expansion estimates was much less than a percent.