Summability of Connected Correlation Functions of Coupled Lattice Fields

Abstract

We consider two nonindependent random fields and defined on a countable set Z. For instance, or , where I denotes a finite set of possible "internal degrees of freedom" such as spin. We prove that, if the cumulants of and enjoy a certain decay property, then all joint cumulants between and are -summable in the precise sense described in the text. The decay assumption for the cumulants of and is a restricted summability condition called -clustering property. One immediate application of the results is given by a stochastic process whose state is -clustering at any time t: then the above estimates can be applied with and 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 -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-Gaussian Wick polynomials and their connection to cumulants.