An argument is presented for a certain asymptotic form for the correlation functions of two local dynamical variables when the system is near its critical point and in the one phase region. If
A(r) and
B(r) are local dynamical variables and $\ensuremath{\omega}$ stands for the variables specifying the thermodynamic state, it is found that for $\ensuremath{\omega}$ near the critical point ${\ensuremath{\omega}}_{c}$ and for
r large enough, $〈A(\mathbf{r})B(0)〉\ensuremath{-}〈A(\mathbf{r})〉〈B(0)〉\ensuremath{\simeq}\frac{{d}_{A}(\ensuremath{\omega}){d}_{B}(\ensuremath{\omega}){e}^{\ensuremath{-}\ensuremath{\kappa}(\ensuremath{\omega})}}{{r}^{b(\ensuremath{\omega})}}$, where $\ensuremath{\kappa}(\ensuremath{\omega})$ and $b(\ensuremath{\omega})$ are the same for a large class of
A and
B and $\ensuremath{\kappa}(\ensuremath{\omega})\ensuremath{\rightarrow}0$, $b(\ensuremath{\omega})\ensuremath{\rightarrow}b\ensuremath{\ge}0 \mathrm{as} \ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}$. For the Ising model of any dimension in zero field and below the critical temperature, the additional assumption of a single correlation length implies the relationship ${\ensuremath{\gamma}}^{\ensuremath{'}}+2\ensuremath{\beta}=2\ensuremath{-}{\ensuremath{\alpha}}^{\ensuremath{'}}$, which is often conjectured for the critical exponents.
There are two independent critical exponents that describe the behavior of systems near their critical point. However, at the critical point only the exponent η, which describes the decay of the correlation function, is usually discussed. We emphasize that there is a second independent exponent η′ that describes the decay of the fourth-order correlation function. The exponent η′ is related to the exponents determining the behavior of thermodynamic functions near criticality via a fluctuation-response equation for the specific heat. We also discuss a scaling law for η′.