The Logarithmic Sobolev Inequality in Infinite dimensions for Unbounded Spin Systems on the Lattice with non Quadratic Interactions

TL;DR

This paper investigates conditions under which the Logarithmic Sobolev Inequality extends to infinite volume Gibbs measures for unbounded spin systems on a one-dimensional lattice with non-quadratic, non-convex interactions.

Contribution

It establishes criteria for extending the Logarithmic Sobolev Inequality to infinite-volume measures in non-quadratic, unbounded spin systems without uniform bounds on second derivatives.

Findings

Log-Sobolev inequality holds under certain boundary conditions.

Extension criteria depend on single-site measure properties.

Applicable to non-convex, unbounded interactions.

Abstract

We are interested in the Logarithmic Sobolev Inequality for the infinite volume Gibbs measure with no quadratic interactions. We consider unbounded spin systems on the one dimensional Lattice with interactions that go beyond the usual strict convexity and without uniform bound on the second derivative. We assume that the one dimensional single-site measure with boundaries satisfies the Log-Sobolev inequality uniformly on the boundary conditions and we determine conditions under which the Log-Sobolev Inequality can be extended to the infinite volume Gibbs measure.