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

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, thatDifferentiating twice at zero gives the infrared bound: for every nonzero torus momentum $k$,up to the harmless normalization convention used for the Fourier transform and Hamiltonian.