Summability of joint cumulants of nonindependent lattice fields

Abstract

We consider two nonindependent random fields $\psi$ and $\varphi$ 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 $\varphi$ are ${\ell }_1$-clustering up to order $2n$, then all joint cumulants between $\psi$ and $\varphi$ 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 $\varphi ={\psi }_0$ 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.

14 pages

References in corpus (3)

• Kinetic Limit for Wave Propagation in a Random Medium
• Wick polynomials and time-evolution of cumulants
• Kinetic theory of phonons in weakly anharmonic particle chains

Cited by in corpus (2)

• Wick polynomials and time-evolution of cumulants
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