Let $\Delta$ be the torus graph Laplacian and let $v:\Lambda_L\to\mathbb R^n$ have zero spatial mean. <Gaussian domination> states, in a standard normalization, that
$$
\frac{Z(v)}{Z(0)}
\leq
\exp\left\{\frac1{2\beta}(v,(-\Delta)^{-1}v)\right\}.
$$
Differentiating twice at zero gives the <infrared bound>: for every nonzero torus momentum $k$,
$$
\widehat G_L(k)
\leq\frac{n}{2\beta\,\varepsilon(k)},
\qquad
\varepsilon(k)=\sum_{j=1}^d(1-\cos k_j),
$$
up to the harmless normalization convention used for the Fourier transform and Hamiltonian.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 209 / 2 / i / Solution

Gaussian domination bounds the partition function of a reflection-positive spin model under a mean-zero external source by the partition function times a Gaussian exponential involving the inverse lattice Laplacian.


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