Extensions of the Brascamp-Lieb Inequality and the Dipole Gas

TL;DR

This paper extends the Brascamp-Lieb inequality to lattice field models and demonstrates that charge-charge correlations in the Coulomb dipole gas are approximately Gaussian, using a novel stochastic dynamics approach.

Contribution

It introduces a new stochastic dynamics method to analyze the dipole gas, extending previous results and providing deeper understanding of charge correlations.

Findings

Charge-charge correlations are close to Gaussian.

New stochastic dynamics approach is effective.

Results extend previous work by Dimock-Hurd and Conlon-Spencer.

Abstract

This paper is concerned with lattice field models in dimension at least 2. The action is a uniformly convex function of the gradient of the field. 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.