Gaussian concentration, integral probability metrics, and coupling functionals for infinite lattice systems

TL;DR

This paper establishes a transport-entropy framework for Gaussian concentration inequalities on infinite product spaces, revealing a duality between integral probability metrics and coupling functionals, and connecting concentration with thermodynamic limits.

Contribution

It introduces a novel duality between integral probability metrics and coupling functionals in infinite lattice systems, extending classical results beyond metric-induced costs.

Findings

Integral probability metric and coupling functional coincide in finite volume.

Marton's coupling inequality is equivalent to Gaussian concentration in all finite volumes.

Metrics converge to the -metric in the thermodynamic limit.

Abstract

We develop a transport-entropy framework for Gaussian concentration inequalities on the infinite product space $S^{\mathbb Z^d}$, where $S$ is a finite set, in which sensitivity is measured by the $\ell^2$-norm of local oscillations. We show that the associated transportation costs cannot be induced by any metric or cost function on the configuration space, due to a structural lack of extensivity in infinite product spaces. Our main result proves that the associated integral probability metric and coupling functional coincide in finite volume, yielding a duality extending the classical Kantorovich-Rubinstein theorem beyond the metric setting. As a consequence, Marton's coupling inequality in all finite volumes is equivalent to Gaussian concentration, yielding a new characterization in the infinite-product setting. In the translation-invariant setting, the corresponding metrics converge in the thermodynamic limit to the $\bar d$-metric. We further introduce a thermodynamic Gaussian concentration bound and prove its equivalence with a transport-entropy inequality involving the relative entropy density.