LAUSR

We consider two nonindependent random fields $$\psi $$ψ and $$\phi $$ϕ defined on a countable set Z. For instance, $$Z=\mathbb {Z}^d$$Z=Zd or $$Z=\mathbb {Z}^d\times I$$Z=Zd×I, where I denotes a finite set of possible “internal degrees of freedom” such as spin. We prove that, if the cumulants of $$\psi $$ψ and $$\phi $$ϕ enjoy a certain decay property, then all joint cumulants between $$\psi $$ψ and $$\phi $$ϕ are $$\ell _2$$ℓ2-summable in the precise sense described in the text. The decay assumption for the cumulants of $$\psi $$ψ and $$\phi $$ϕ is a restricted $$ \ell _1$$ℓ1 summability condition called $$\ell _1$$ℓ1-clustering property. One immediate application of the results is given by a stochastic process $$\psi _t(x)$$ψt(x) whose state is $$\ell _1$$ℓ1-clustering at any time t: then the above estimates can be applied with $$\psi =\psi _t$$ψ=ψt and $$\phi =\psi _0$$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 $$\ell _1$$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-Gaussian Wick polynomials and their connection to cumulants