Translational invariance makes the pointwise <variance> independent of $\mathbf r$. Applying <Parseval identity> to the stated Fourier convention gives
$$
\sigma^2
=\langle|\phi(\mathbf r)|^2\rangle_0
=\frac1V\sum_{\mathbf q}\langle|\phi_{\mathbf q}|^2\rangle_0
=\frac1V\sum_{\mathbf q}\frac1{J(q)}.
$$
In the <thermodynamic limit>, the reciprocal-lattice sum becomes
$$
\sigma^2
=\int\frac{d^d\mathbf q}{(2\pi)^d}\frac1{J(q)}.
$$
At each point $\phi(\mathbf r)$ is a centered <Gaussian random variable>, so the supplied absolute third moment gives
$$
g\int d^d\mathbf r\,\langle|\phi(\mathbf r)|^3\rangle_0
=\frac{4gV}{\sqrt{2\pi}}\sigma^3
=\frac{4gV}{\sqrt{2\pi}}
\left[
\int\frac{d^d\mathbf q}{(2\pi)^d}\frac1{J(q)}
\right]^{3/2}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 344 / 1 / e / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-344/1/e/solution