An Inequality for Gaussians on Lattices

$ \newcommand{\R}{\ensuremath{$\mathbb{R}$}} \newcommand{\lat}{$\mathcal{L}$} \newcommand{\ensuremath}[1]{#1} $Weshowthatforanylattice$\lat \subseteq \R^n$andvectors$\vec{x}, \vec{y} \in \R^n$, $$\rho (\text{lat}+\vec{x}{)}^2\rho (\text{lat}+\vec{y}{)}^2\le \rho (\text{lat}{)}^2\rho (\text{lat}+\vec{x}+\vec{y})\rho (\text{lat}+\vec{x}-\vec{y}),$$ where $\rho$ is the Gaussian measure $\rho (A):={\sum }_{\vec{w}\in A}\exp (-\pi \mathrm{\parallel }\vec{w}{\mathrm{\parallel }}^2)$. 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.