Summability of Connected Correlation Functions of Coupled Lattice Fields

We consider two nonindependent random fields (psi ) and (phi ) defined on a countable set Z. For instance, (Z=mathbb {Z}^d) or (Z=mathbb {Z}^dtimes I), where I denotes a finite set of possible “internal degrees of freedom” such as spin. We prove that, if the cumulants of (psi ) and (phi ) enjoy a certain decay property, then all joint cumulants between (psi ) and (phi ) are (ell _2)-summable in the precise sense described in the text. The decay assumption for the cumulants of (psi ) and (phi ) is a restricted ( ell _1) summability condition called (ell _1)-clustering property. One immediate application of the results is given by a stochastic process (psi _t(x)) whose state is (ell _1)-clustering at any time t: then the above estimates can be applied with (psi =psi _t) and (phi =psi _0) 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 (ell _1)-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