Invariances in variance estimates

TL;DR

This paper enhances the Brascamp-Lieb variance inequality by incorporating invariance properties, leading to improved spectral gap estimates for symmetric log-concave measures and spin systems.

Contribution

It introduces variants of the inequality that account for measure invariances, advancing spectral gap analysis in symmetric settings.

Findings

Improved spectral gap estimates for symmetric log-concave measures

Enhanced variance bounds for spin systems with symmetries

New variants of the Brascamp-Lieb inequality incorporating invariance

Abstract

We provide variants and improvements of the Brascamp-Lieb variance inequality which take into account the invariance properties of the underlying measure. This is applied to spectral gap estimates for log-concave measures with many symmetries and to non-interacting conservative spin systems.