Hypercontractivity for spin systems of infinite extension (Q1346966)

The author considers a spin system on \(Z^ d\), with values in a compact connected Riemannian manifold of finite dimension. He proves the equivalence between first a weak mixing condition, second the control of the spectral gap and third the control of logarithmic Sobolev constants, for not necessarily finite range Gibbs potentials. Consequently, he obtains informations about the \(L^ 2\)-decay of semigroups to the equilibrium and the decorrelation decay. There is no transitory rate between an algebraic decay as \(t^{-2d}\) (resp. \(| j - k|^{- 2d}\)), and an exponential decay. In a last part, the author considers discrete compact spins and proves the same equivalence result between the three above properties. In this particular situation, he deduces moreover that in case of phase transition (several Gibbs measures), the correlation between two spins at locations \(j\) and \(k\) cannot decrease uniformly more quickly than \(| j - k |^{-2d}\).