Summability of Connected Correlation Functions of Coupled Lattice Fields.

We consider two nonindependent random fields Ψ and Φ defined on a countable set Z. For instance, Z = Zd or Z = Zd X I, where I denotes a finite set of possible "internal degrees of freedom" such as spin.We prove that, if the cumulants of Ψ and Φenjoy a certain decay property, then all joint cumulants between Ψ and Φ are ℓ2-summable in the precise sense described in the text. The decay assumption for the cumulants of Ψ and Φ is a restricted ℓ1 summability condition called ℓ1-clustering property. One immediate application of the results is given by a stochastic process Ψt (x) whose state is ℓΨ = Ψt and Φ= Ψ0 and we obtain uniform in t estimates for the summability of time-correlations of the field. The above clustering assumption is obviously satisfied by any ℓ1-clustering stationary state of the process and our original motivation for the control of the summability of time-correlations comes from a quest for a rigorous control of the Green-Kubo correlation function in such a system. A key role in the proof is played by the properties of non-GaussianWick polynomials and their connection to cumulants. [ABSTRACT FROM AUTHOR]