# Equivalence of decay of correlations, the log-Sobolev inequality, and of the spectral gap

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Georg Menz, Stanford University
Fine Hall 322

In this talk we consider a lattice system of unbounded real-valued spins, which is described by its Gibbs measure mu. We discuss how a known result for finite-range interaction is generalized to infinite-range. The result states that it is equivalent: The correlations of the Gibbs measure mu decay, the Gibbs measure mu satisfies a log-Sobolev inequality uniformly in the systems size and the boundary values, and mu satisfies a uniform spectral gap. Such a statement is interesting, because it connects a static property of the equilibrium state mu of the system to a dynamic property of the system i.e. how fast the Glauber dynamics converges to equilibrium.