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

We investigate unbounded continuous spin-systems with infinite-range interactions. We develop a new technique for deducing decay of correlations from a uniform Poincaré inequality based on a directional Poincaré 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.