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For dimensionless scalar Landau-Ginzburg theory with quadratic coefficient $r=a_{t}t$, gradient coefficient $c>0$ and quartic term $u{\varphi }^4/4$, the correlation-volume derivation of the scalar Ginzburg ratio gives $\mathcal{G}\sim u{c}^{-d/2}{a}_t^{(d-4)/2}|t{|}^{-(4-d)/2}$. For $d<4$, solving $\mathcal{G}\sim 1$ gives the displayed nonuniversal window. Mean-field self-consistency requires $|t|\gg \mathrm{Gi}$ while still staying in the domain of the local expansion. In three dimensions $\mathrm{Gi}\sim u^2/(c^3{a}_t)$. Numerical prefactors depend on the averaging convention and microscopic normalization; no universal material value follows from dimensional power counting. At four dimensions use the marginal Ginzburg criterion instead of this singular exponent formula.

The Ginzburg criterion tests whether fluctuations on the scale of one correlation volume are small compared with the squared equilibrium order parameter. In the scalar quartic Landau-Ginzburg theory of part (a), the ordered mean-field value is $M^2=|r|/u$, and its longitudinal correlation length is proportional to $(c/|r|{)}^{1/2}$.

Average the fluctuation field over a region of size $\xi$. Its smoothing factor selects momenta of order ${\xi }^{-1}$ or smaller. The Gaussian correlation function then gives, up to a positive dimension-dependent constant,The last step follows directly by setting $q=\sqrt{|r|/c}k$. Dividing by $M^2$ proves the correlation-volume derivation of the scalar Ginzburg ratio:For $d<4$, the ratio diverges as the transition is approached, so there is a fluctuation-dominated critical region of a phase transition in which mean-field critical exponents cannot be trusted. With $r=r_{0}t$, an estimate of its width isin fixed microscopic units. Its numerical coefficient depends on the averaging convention and microscopic normalization.

For $d>4$, the ratio tends to zero and the long-distance mean-field approximation becomes self-consistent. Four is therefore the ordinary quartic upper critical dimension. At $d=4$ this test is marginal: its thermal power is zero, and the running quartic interaction produces logarithmic corrections. The marginal Ginzburg criterion alone cannot decide those logarithms. Short-distance contributions to an unaveraged variance renormalize the coefficients; they should not be mistaken for the long-wavelength fluctuations used in this criterion. Nor does the criterion by itself prove that a proposed ordered phase exists below its lower critical dimension.