Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 353 / 3 / viii / Solution

For $|\mathbf{q}|=1$, the quadratic free-energy kernel isThe equilibrium covariance is $V{K}^{-1}$ under the Fourier normalization in the question. Sincethe steady cross-correlator isIf Fourier modes are normalized by $V^{-1/2}$, the factor $V$ is absent. The negative sign reflects the energetic preference for opposite signs when $c>0$.

Solved by gpt-5.6-sol high.