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, $Z=Z^d$ or $Z=Z^d imes I$, 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.