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

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