= Solution
The <hyperscaling relation> expresses the idea that the singular dimensionless <free energy> in one <correlation volume> is of order one. A macroscopic volume $V$ contains of order $V/\xi^d$ such correlated regions, so
$$
f_s(t,h)\asymp\xi(t,h)^{-d}.
$$
Substitution of the homogeneous <correlation length> gives
$$
\xi^{-d}=|t|^{d\nu}[g_{\xi,\pm}(h/|t|^\Delta)]^{-d}.
$$
This has exactly the required homogeneous field variable. Comparing its thermal power with that of the singular <free-energy density> proves
$$
\boxed{d\nu=2-\alpha.}
$$
The argument concerns powers, not equality of the two scaling functions or their amplitudes. It assumes the <correlation length> supplies the only singular length scale and that no <dangerously irrelevant coupling> changes the free energy per correlated region. For ordinary short-range quartic theory, <hyperscaling> holds below the <upper critical dimension> four, with the usual qualifications for marginal logarithms at four. Above four, <mean-field critical exponents> $\alpha=0$, $\nu=1/2$ do not satisfy $d\nu=2-\alpha$; the quartic coupling is dangerously irrelevant. The stated homogeneous form of $\xi$ alone therefore does not prove hyperscaling without the correlation-volume assumption.
Back to article page
