Log-Sobolev inequalities for infinite-dimensional Gibbs measures with non-quadratic interactions

TL;DR

This paper establishes conditions under which infinite-dimensional Gibbs measures with non-quadratic interactions satisfy the log-Sobolev inequality, extending known results from quadratic to higher-order interactions.

Contribution

It proves that uniform log-Sobolev inequalities for single-site measures imply the same for the entire Gibbs measure with higher-order interactions.

Findings

Infinite-dimensional Gibbs measures satisfy log-Sobolev inequalities under certain conditions.

Uniformity in single-site measures leads to global inequalities.

Extension from quadratic to non-quadratic interactions.

Abstract

We focus on the log-Sobolev inequality for spin systems on the lattice with interactions of higher order than quadratic. We show that if the one-dimensional single-site measure with boundaries satisfies the log-Sobolev inequality uniformly in the boundary conditions then the infinite-dimensional Gibbs measure also satisfies the inequality under appropriate conditions on the phase and the interactions.