= Solution
For a centered <Gaussian random field>, <Wick theorem> takes the classical form of the <Isserlis theorem>: every even correlator is a sum over pairings of two-point correlators, and every odd correlator vanishes. For example, with $G_i=\zeta_G(\mathbf x_i)$,
$$
\langle G_1G_2G_3G_4\rangle=\langle G_1G_2\rangle\langle G_3G_4\rangle+\langle G_1G_3\rangle\langle G_2G_4\rangle+\langle G_1G_4\rangle\langle G_2G_3\rangle.
$$
A $2n$-point correlator has $(2n-1)!!$ pairings. Equivalently, all <connected correlation functions> beyond order two vanish. \b[The <covariance> completely determines the centered Gaussian statistics; higher ordinary correlators can be nonzero, but contain no independent connected information.] For a field with nonzero mean, apply these statements to the centered field and then restore its mean.
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