# Summability of Connected Correlation Functions of Coupled Lattice Fields

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## 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 \(\psi \) 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

## Keywords

Cumulants Decorrelation estimates Equilibrium time-correlations BBGKY hierarchy Wick polynomials Dynamics of lattice fields Discrete non-linear Schrodinger equation## Notes

### Acknowledgements

We thank Thierry Bodineau, Antti Kupiainen, Sergio Simonella, and Herbert Spohn for useful discussions on the topic. The work has been supported by the Academy of Finland via the Centre of Excellence in Analysis and Dynamics Research (Projects 271983 and 307333) and from an Academy Project (Project 258302). A. Nota also acknowledges support from the German Science Foundation (DFG) via the project CRC 1060 “The mathematics of emergent effects.” The work and the related discussions have partially occurred in workshops supported by the French Ministry of Education through the grant ANR (EDNHS), by the Institut Henri Poincaré—Centre Émile Borel, Paris, France, during the trimester “Stochastic Dynamics Out of Equilibrium”, and by the Erwin Schrödinger Institute (ESI), Vienna, Austria.

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