Summability of Connected Correlation Functions of Coupled Lattice Fields

Abstract

We consider two nonindependent random fields $\psi$ and $\phi$ defined on a countable set $Z$. For instance, $Z={\mathbb Z}^d$ or $Z={\mathbb Z}^d\times I$, where $I$ denotes a finite set of possible "internal degrees of freedom" such as spin. We prove that, if the cumulants of both $\psi$ and $\phi$ are $\ell_1$-clustering up to order $2 n$, then all joint cumulants between $\psi$ and $\phi$ are $\ell_2$-summable up to order $n$, in the precise sense described in the text. We also provide explicit estimates in terms of the related $\ell_1$-clustering norms, and derive a weighted $\ell_2$-summation property of the joint cumulants if the fields are merely $\ell_2$-clustering. 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.