= Solution
The <renormalization group> compares descriptions at successively longer length scales. First eliminate short-wavelength degrees of freedom, then rescale coordinates and fields to restore the reference cutoff. This changes the couplings, including interactions and additive free-energy terms generated during elimination. A <renormalization-group fixed point> is a scale-invariant description. Linearized directions growing under coarse graining are relevant, shrinking directions are irrelevant, and zero-eigenvalue directions are marginal and require higher-order analysis.
A diverging <correlation length> means that many coarse-graining steps can approach a <renormalization-group fixed point> before a relevant perturbation becomes large. Systems with different microscopic details but the same <renormalization-group fixed point> and relevant directions share universal critical behavior. For the quadratic model below, <Fourier modes> decouple, so eliminating a high-momentum shell produces a Gaussian <determinant> but does not couple the remaining modes. The anisotropic <gradient> kernel requires different length rescalings in the parallel and transverse directions. This is an <anisotropic renormalization group>, not an isotropic rescaling of all $d$ coordinates.
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