Extensions of the Brascamp-Lieb Inequality and the Dipole Gas

Abstract

This paper is concerned with \(d\ge 2\) lattice field models with action \(V(\nabla \phi (\cdot ))\), where \(V:\mathbb {R}^d\rightarrow \mathbb {R}\) is a uniformly convex function. The main result Theorem 1.4 proves that charge-charge correlations in the Coulomb dipole gas are close to Gaussian. These results go beyond previous results of Dimock-Hurd and Conlon-Spencer. The approach in the paper is based on the observation that the sine-Gordon probability measure corresponding to the dipole gas is the invariant measure for a certain stochastic dynamics. The stochastic dynamics here differs from the stochastic dynamics in previous work used to study the problem.