HYPERCONTRACTIVITY FOR SPIN SYSTEMS WITH INFINITE-RANGE

We consider a spin system on Z(d). We prove the equivalence between first a weak mixing condition, secondly the controle of spectral gap and thirdly the controle of logarithmic Sobolev constants for non necessarily finite range Gibbs potentials. Hence we draw consequences concerning the L(2) decay to equilibrium and the correlations decay : there is no transitory rate between an algebraic decay t(-2d) (resp. \j - k\(-2d)) and exponential decay. The general results are obtained as for both continuous and discrete compact spins.