 Solution

After separating an additive identity term and using dimensionless couplings, let $R$ be one blocking-and-rescaling step. A renormalization-group fixed point satisfies $R(u_{\ast })=u_{\ast }$. Near it, linearization gives scaling coordinates $w_i$ withAt a critical renormalization-group fixed point, no finite characteristic length survives. Since the correlation length measured in lattice units transforms as $\xi (Ru)=\xi (u)/b$, a nonzero scale-invariant critical length must be infinite. A sequence of systems approaching the same fixed point therefore has common long-distance behavior.

The scaling hypothesis for critical phenomena asserts that the singular free-energy density transforms homogeneously under these scaling coordinates. It follows from repeated blocking near the fixed point, once analytic backgrounds are removed and no dangerously irrelevant coupling supplies an additional singular dependence. The temperature and magnetic-field scaling directions set the principal departures from criticality; irrelevant directions label microscopic differences that disappear at long distances.