Spectral Gap Inequalities on Nilpotent Lie Groups in Infinite Dimensions

TL;DR

This paper establishes a general framework for spectral gap inequalities for Gibbs measures on infinite-dimensional spin spaces over nilpotent Lie groups, extending existing results using weak bounds and inequalities.

Contribution

It introduces a novel framework for spectral gap inequalities in infinite-dimensional settings over nilpotent Lie groups, with new conditions and examples.

Findings

Framework applicable to Gibbs measures on nilpotent Lie groups

Provides sufficient conditions for spectral gap inequalities

Includes examples using Kaplan norm and Carnot-Caratheodory distance

Abstract

We develop a general framework for spectral gap inequalities for Gibbs measures on infinite dimensional spin spaces over nilpotent Lie groups in terms of weak U-bounds and weak single-site spectral gap inequalities. We then provide sufficient conditions on the local specification and give examples of measures constructed using the Kaplan norm and generalising a few results for the Carnot-Caratheodory distance on the Heisenberg group.