Equivalence of decay of correlations, log-Sobolev inequalities, and Poincare inequalities in spin systems with infinite range interactions

Christopher Henderson, Georg Menz

Published 2014-10-15Version 1

Yoshida proved in the setting of unbounded continuous spins with finite-range interactions the equivalence of a uniform log-Sobolev inequality, a uniform Poincar\'e inequality, and a suitable decay of spin- spin correlations. We obtain analogous results in the case of infinite-range interactions which decay only algebraically. The results also hold under less restricting conditions on the single-site potentials of the Hamiltonian. The main new technique is to get decay of correlation with the help of a directional Poincar\'e inequality deduced by an averaging technique. In addition, we show that in the case of ferromagnetic interaction a weaker algebraic decay of correlations can be bootstrapped into a decay of the same order as the spin-spin interactions.