The log-Sobolev inequality with quadratic interactions

TL;DR

This paper proves that infinite-dimensional Gibbs measures with quadratic interactions inherit the log-Sobolev inequality from their single-site measures, even beyond convexity at infinity, expanding understanding of measure stability under interactions.

Contribution

It establishes that quadratic interactions preserve the log-Sobolev inequality in infinite-dimensional Gibbs measures, including non-convex phase examples.

Findings

Gibbs measures with quadratic interactions satisfy log-Sobolev inequalities.

The result applies even when the phase is non-convex at infinity.

Examples demonstrate the inequality beyond convex phases.

Abstract

We assume one site measures without a boundary $e^{-\phi(x)}dx/Z$ that satisfy a log-Sobolev inequality. We prove that if these measures are perturbed with quadratic interactions, then the associated infinite dimensional Gibbs measure on the lattice always satisfies a log-Sobolev inequality. Furthermore, we present examples of measures that satisfy the inequality with a phase that goes beyond convexity at infinity.