= Solution
Use the <connected correlation function> $G(x)=\langle M(x)M(0)\rangle-\langle M\rangle^2$ in a selected <pure thermodynamic phase>. The exponential <correlation length> is defined by $\xi^{-1}=-\lim_{|x|\to\infty}|x|^{-1}\log|G(x)|$, when this limit exists in the chosen direction. Linearizing the <Landau-Ginzburg theory> around a stable uniform minimum $M_0$ gives
$$
\mathcal F^{(2)}=\frac12\int d^Dx\,[K|\nabla\delta M|^2+V''(M_0)(\delta M)^2],\qquad
\widetilde G(q)=\frac{k_BT}{Kq^2+V''(M_0)}.
$$
Hence the <Landau scalar correlation length> is $\boxed{\xi=\sqrt{K/V''(M_0)}}$. Away from criticality the <massive Gaussian field correlation tail> behaves as $|x|^{-(D-1)/2}e^{-|x|/\xi}$, up to its amplitude. At a continuous transition $V''$ tends to zero, so $\xi$ diverges and fields vary on scales much longer than the lattice spacing. This motivates a <gradient> expansion, but the simultaneous growth of correlated fluctuations is precisely why the <Ginzburg criterion> must be checked.
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