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 $B{|t|}^{\ensuremath{\beta}}$, ${{A}_{F}}^{\ifmmode\pm\else\textpm\fi{}}{|t|}^{2\ensuremath{-}\ensuremath{\alpha}}$, ${C}^{\ifmmode\pm\else\textpm\fi{}}{|t|}^{\ensuremath{-}\ensuremath{\gamma}}$, and ${\ensuremath{\xi}}_{0}{|t|}^{\ensuremath{-}\ensuremath{\nu}}$ [where $t=\frac{({p}_{c}\ensuremath{-}p)}{{p}_{c}}$,
p being the concentration of nonzero bonds, and +(-) stands for
p<pc (
p>pc)] 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 $\frac{{{A}_{F}}^{+}}{{{A}_{F}}^{\ensuremath{-}}}$, $\frac{{C}^{+}}{{C}^{\ensuremath{-}}}$, ${{A}_{F}}^{+}{B}^{\ensuremath{-}2}{C}^{+}$, $\frac{{\ensuremath{\xi}}_{0}^{+}}{{\ensuremath{\xi}}_{0}^{\ensuremath{-}}}$, and ${{A}_{F}}^{+}{({\ensuremath{\xi}}_{0}^{+})}^{d}$ (
d is the dimensionality) are universal. Similar quantities are found for the behavior at
p=pc (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 $\ensuremath{\epsilon}=6\ensuremath{-}d$. 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, when logarithmic corrections appear. Additional universal ratios are obtained for the confluent correction to scaling terms.
Thermodynamic functions in the vicinity of an ordinary critical point are expected to obey asymptotic scaling laws as $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ and the ordering field,
h, approach zero. However, the optimal scaling variables are the nonlinear scaling fields, ${g}_{t}=t+{b}_{t}{h}^{2}+{c}_{t}{t}^{2}+\ensuremath{\cdots}$ and ${g}_{h}=h(1+{c}_{h}t+\ensuremath{\cdots})$. The nonlinearities yield correction factors to the leading power-law (and scaling) variation of thermodynamic quantities,
L, of the form ($1+{a}_{L}t+{b}_{L}{t}^{2}+\ensuremath{\cdots}$), etc., where the correction amplitudes
aL,bL,…, are uniquely determined by the nonlinear scaling-field coefficients. It follows that "analytic" corrections to, e.g., the susceptibility, are directly related to those for the free energy and magnetization (in zero field). The term
bth2 also generates nonanalytic contributions such as an additive, energylike term, varying as ${|t|}^{1\ensuremath{-}\ensuremath{\alpha}}$ in the zero-field susceptibility, and factors like ($1+{c}_{L}{|h|}^{2\ensuremath{-}\frac{1}{\ensuremath{\Delta}}}$) on the critical isotherm,
t=0. Irrelevant scaling fields yield further, in general distinct, nonanalytic corrections, and cause shifts in
Tc and various amplitudes although "universal" ratios remain constant.
The thermodynamic functions of an
n-component
d-dimensional ${\ensuremath{\sigma}}^{4}$ theory are written in the form ${f}_{i}={A}_{i}{|t|}^{\ensuremath{-}{\ensuremath{\lambda}}_{i}}(1+{a}_{i}{|t|}^{\ensuremath{\Delta}})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$. The correction-to-scaling amplitudes
ai are calculated to leading order in
n1 for
2<d<4, and the universality of their ratios is explicitly shown. The effective exponents ${\ensuremath{\lambda}}_{i,\mathrm{eff}}={\ensuremath{\lambda}}_{i}\ensuremath{-}{a}_{i}\ensuremath{\Delta}{|t|}^{\ensuremath{\Delta}}$ violate the thermodynamic scaling laws already at this leading order. The amplitudes
ai have singularities (as functions of
d) at the dimensionalities ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}_{l}=2+\frac{4}{l},l=3,4,5,\dots{}$. Above
Tc, the specific heat and the susceptibility correction amplitudes
ac+ and ${a}_{\ensuremath{\chi}}^{+}$ are discontinuous at ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}_{l}$ and at ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}_{2l}(l\ensuremath{\ge}2)$, respectively. Below
Tc, the magnetization and the stiffness correction amplitudes are zero, ${a}_{M}={a}_{{\ensuremath{\rho}}_{s}}=0$, while ${a}_{c}^{\ensuremath{-}}$ diverges at ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}_{2l+1}(l\ensuremath{\ge}1)$ but is continuous at
d2l (in particular at
d=3). Results are also given for universal ratios among corrections-to-scaling amplitudes for the dependence of thermo-dynamic functions on the magnetization at
T=Tc and at a nonzero field.
Correction-to-scaling amplitudes are calculated to second order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ by renormalized perturbation theory. It is shown that ratio of any two such amplitudes is universal, and that thermodynamic scaling laws among effective exponents break down beyond zeroth order in $\ensuremath{\epsilon}$. Values found are $\frac{{{a}_{\ensuremath{\xi}}}^{+}}{{{a}_{\ensuremath{\chi}}}^{+}}=0.65 (n=1, d=3) \mathrm{and} 0.63 (n=2, d=3)$ in good agreement with high-temperature series, and $\frac{{{a}_{m}}^{\ensuremath{-}}}{{{a}_{\ensuremath{\chi}}}^{+}}=0.85 (n=1, d=3)$.
We estimate, at
d=3, from ${\ensuremath{\varphi}}^{4}$ field theory the universal ratios of confluent critical correction amplitudes relative to the correlation length, the susceptibility, and the specific heat above
Tc. Our central values are comparable to these available from analysis of high-temperature series and the $\ensuremath{\epsilon}$ expansion. Furthermore, experiments in progress on a binary mixture are in good agreement with one of our results, the ratio of the confluent corrections for the specific heat and susceptibility which have not been previously estimated. We also present, in detail, a derivation of the universality of these quantities in the framework of field theory.