Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 72 / 2 / c

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, soSubstitution of the homogeneous correlation length givesThis has exactly the required homogeneous field variable. Comparing its thermal power with that of the singular free-energy density provesThe 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.