Equivalence of a mixing condition and the LSI in spin systems with infinite range interaction

TL;DR

This paper establishes an equivalence between a mixing condition and the Logarithmic Sobolev Inequality (LSI) in infinite-range spin systems, introducing a new technique based on directional Poincaré inequalities.

Contribution

It develops a novel averaging-based method to derive decay of correlations from a uniform Poincaré inequality and proves its equivalence to the Dobrushin-Shlosman mixing condition in such systems.

Findings

Decay of correlations is equivalent to the Dobrushin-Shlosman mixing condition.

A new averaging technique for directional Poincaré inequalities is introduced.

Partial results on the relationship between relaxation rates in ferromagnetic and non-ferromagnetic systems.

Abstract

We investigate unbounded continuous spin-systems with infinite-range interactions. We develop a new technique for deducing decay of correlations from a uniform Poincar\'e inequality based on a directional Poincar\'e inequality, which we derive through an averaging procedure. We show that this decay of correlations is equivalent to the Dobrushin-Shlosman mixing condition. With this, we also state and provide a partial answer to a conjecture regarding the relationship between the relaxation rates of non-ferromagnetic and ferromagnetic systems. Finally, we show that for a symmetric, ferromagnetic system with zero boundary conditions, a weaker decay of correlations can be bootstrapped.