Abstract
The matching of leading singular contributions to the different mean-field or critical behaviors of a system with long-but-finite-ranged spatially anisotropic interactions leads to "scaling laws" connecting criticial exponents for different dimensions. The results are exact for mean-field theory (MFT) and spherical model exponents and predict MFT exponents for $d=4$. From the known $d=2$ Ising-model exponents, the scaling laws give $\ensuremath{\alpha}=0$, $\ensuremath{\beta}=\frac{4}{11}$, $\ensuremath{\gamma}=\frac{14}{11}$, $\ensuremath{\nu}=\frac{2}{3}$ for $d=3$. Agreement with previous results for the $d=2,3$ excluded volume problem is also quite good. For the $X\ensuremath{-}Y$ and Heisenberg models the results predict $\ensuremath{\gamma}\ensuremath{\approx}2.0 \mathrm{and} 2.4$, respectively, for $d=2$.
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