An Inequality for Gaussians on Lattices

$ \newcommand{\R}{\ensuremath{\mathbb{R}}} \newcommand{\lat}{\mathcal{L}} \newcommand{\ensuremath}[1]{#1} $We show that for any lattice $\lat \subseteq \R^n$ and vectors $\vec{x}, \vec{y} \in \R^n$, \[ \rho(\lat + \vec{x})^2 \rho(\lat + \vec{y})^2 \leq \rho(\lat)^2 \rho(\lat + \vec{x} + \vec{y}) \rho(\lat + \vec{x} - \vec{y}) \; , \] where $\rho$ is the Gaussian measure $\rho(A) := \sum_{\vec{w} \in A} \exp(-\pi \| \vec{w} \|^2)$. We show a number of applications, including bounds on the moments of the discrete Gaussian distribution, various monotonicity properties of the heat kernel on flat tori, and a positive correlation inequality for Gaussian measures on lattices.