Solution
= Solution
If the singular free-energy density is set by one correlated region, then $f_s\sim\xi^{-d}$. With reduced temperature $t$, the definitions $\xi\sim|t|^{-\nu}$ and $f_s\sim|t|^{2-\alpha}$ therefore give
$$
|t|^{2-\alpha}\sim|t|^{d\nu},
\qquad
\boxed{2-\alpha=d\nu}.
$$
This is the <hyperscaling relation>; it assumes that no <dangerously irrelevant coupling> introduces an additional scale.