Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 209 / 1 / i / Solution

For a finite graph $\Lambda =(V,E)$ with free boundary conditions, write ${\sigma }_x=(\cos {\theta }_x,\sin {\theta }_x)\in S^1$. The ferromagnetic O(2) model isThe Ginibre inequality says, in particular, that for $a,b\in {\mathbb{Z}}_{\ge 0}^V$,

For the proof, take two independent replicas $\theta ,{\theta }^{\prime }$ and write the covariance as one half of the expectation ofSet $u=(\theta +{\theta }^{\prime })/2$ and $v=(\theta -{\theta }^{\prime })/2$. Product-to-sum identities turn each difference into $-2\sin (a\cdot u)\sin (a\cdot v)$, while every replicated interaction becomesExpand every exponential in a power series and then every cosine power into Fourier modes. Integration over each angle kills all unmatched modes. Because the couplings, field, and entries of $a,b$ are nonnegative, every surviving paired coefficient in the covariance is nonnegative. Their sum is therefore nonnegative, proving the inequality. The same replica expansion proves the usual product version.