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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

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Notes

  1. If one has distinct labels in \(J'_n\) and \(J_n\), achievable always by relabelling of one of the sets, one can safely take here \(J'_n+ J_n=J'_n\cup J_n\), \(\fancyscript{P}(J'_n+ J_n)\) equal to the ordinary partitions of the set \(J'_n\cup J_n\), and also \(S|J_n = S\cap J_n\). Here, we have opted to use the definitions from [18] instead of explicit relabellings.

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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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Correspondence to Jani Lukkarinen.

Appendix: Cumulants and Wick Polynomials

Appendix: Cumulants and Wick Polynomials

We have collected in this Appendix the main results relating to Wick polynomials which were used in the text. The goal is not to give an exhaustive exposition of the topic, but rather to list the minimal amount of definitions and properties needed for the proofs here. In the Gaussian case, one can also identify the Wick polynomials as arising from an orthogonalization procedure, with applications in Wiener chaos expansions and Malliavin calculus [13]. The present general, non-Gaussian case is not directly connected to orthogonal polynomials, and leads to more complicated combinatorial expansions: we refer to [21] for a recent review from the point of view of probability theory and for more results on how graphical representations may be used to facilitate the analysis involving Wick polynomials.

Let us consider a collection \(y_j\), \(j\in J\) where J is some fixed nonempty index set of real or complex random variables on some probability space \((\Omega ,\fancyscript{M},\mu )\). Then for any sequence of indices, \(I=(i_1,i_2,\ldots ,i_n)\in J^n\), we use the following shorthand notations to label monomials of the above random variables:

$$\begin{aligned} y^I = y_{i_1} y_{i_2}\cdots y_{i_n} = \prod _{k=1}^n y_{i_k} \, ,\qquad y^\emptyset := 1\quad \text {if}\quad I=\emptyset . \, \end{aligned}$$
(A.1)

As already explained in an earlier footnote, we consider sequences of indices and not sets of indices in order to avoid more cumbersome notations involving relabelling of elements whenever an element is repeated in a sequence. We will however continue to use set-like notations for subsequences and partitions of sequences. To be precise, these notations are valid only after one has added a unique label for each element of the sequence. For this, we introduce a collection \(\fancyscript{I}\) which consists of those finite subsets \(A\subset \mathbb {N}\times J\) with the property that if \((n,j), (n',j')\in A\) and \((n,j)\ne (n',j')\) then \(n\ne n'\). The empty sequence is identified with \(\emptyset \in \fancyscript{I}\). For nonempty sets, the natural number in the first component serves as a distinct label for each member in A and their order determines the order of the elements in the sequence.

For any \(I\in \fancyscript{I}\) we denote the corresponding moment by \( {\mathbb E}[y^I]\), and the related cumulant by

$$\begin{aligned}&\kappa [y_I] = \kappa _\mu [y_I] = \kappa [y_{i_1}, y_{i_2}, \ldots , y_{i_n}]. \end{aligned}$$
(A.2)

The corresponding Wick polynomial is denoted by

$$\begin{aligned} \mathop {:}\nolimits \!y^I\!\mathop {:}\nolimits = \mathop {:}\nolimits \!y^I\!\mathop {:}\nolimits _\mu =\mathop {:}\nolimits \!y_{i_1} y_{i_2} \ldots y_{i_n}\!\mathop {:}\nolimits . \end{aligned}$$
(A.3)

Both \(\kappa [y^I]\) and \( \mathop {:}\nolimits \!y^I\!\mathop {:}\nolimits \) can be defined recursively if \(I\in \fancyscript{I}\) is such that \({\mathbb E}[|y^E|]<\infty \) for all \(E\subset I\) (see [18]). Explicitly, it suffices to require that

$$\begin{aligned} \mathop {:}\nolimits \!y^{I}\!\mathop {:}\nolimits = y^I - \sum _{E\subsetneq I} {\mathbb E}[y^{I\setminus E}]\, \mathop {:}\nolimits \!y^E\!\mathop {:}\nolimits , \end{aligned}$$
(A.4)

and, choosing some \(x\in I\),

$$\begin{aligned} \kappa [y_I] = {\mathbb E}[y^{I}] - \sum _{E:x\in E\subsetneq I} {\mathbb E}[y^{I\setminus E}] \kappa [y_E]. \end{aligned}$$
(A.5)

Let us also recall that both cumulants and Wick polynomials are multilinear and permutation invariant.

If the random variables \(y_j\), \(j=1,2,\ldots ,n\), have joint exponential moments, then moments, cumulants and Wick polynomials can also be easily generated by differentiation of their respective generating functions which are

$$\begin{aligned} G_{\mathrm {m}}(\lambda ) := {\mathbb E}[\mathrm{{e}}^{\lambda \cdot x}], \quad g_{\mathrm {c}}(\lambda ) := \ln G_{\mathrm {m}}(\lambda ) \quad \text {and}\quad G_{\mathrm {w}}(\lambda ; y) := \frac{\mathrm{{e}}^{\lambda \cdot y}}{{\mathbb E}[\mathrm{{e}}^{\lambda \cdot x}]} = \mathrm{{e}}^{\lambda \cdot y-g_{\mathrm {c}}(\lambda )}. \end{aligned}$$
(A.6)

By evaluation of the I-th derivative at zero, we have

$$\begin{aligned} {\mathbb E}[y^I] = \partial ^I_\lambda G_{\mathrm {m}}(0), \quad \kappa [y_I]= \partial ^I_\lambda g_{\mathrm {c}}(0) \quad \text {and}\quad \mathop {:}\nolimits \!y^I\!\mathop {:}\nolimits = \partial ^I_\lambda G_{\mathrm {w}}(0;y), \end{aligned}$$
(A.7)

where “\(\partial ^I_\lambda \)” is a shorthand notation for \(\partial _{\lambda _{i_1}}\partial _{\lambda _{i_2}}\ldots \partial _{\lambda _{i_n}}\).

It is remarkable that expectations of products of Wick polynomials can be expanded in terms of cumulants, merely cancelling some terms from the standard moment-to-cumulants expansion. The following result, proven as Proposition 3.8 in [18], details the result using the above notations:

Proposition 5.1

Assume that the measure \(\mu \) has all moments of order N, i.e., suppose that \({\mathbb E}[|y^I|]<\infty \) for all \(I\in \fancyscript{I}\) with \(|I|\le N\). Suppose \(L\ge 1\) is given and consider a collection of \(L+1\) index sequences \(J',J_\ell \in \fancyscript{I}\), \(\ell =1,\ldots ,L\), such that \(|J'|+\sum _\ell |J_\ell |\le N\). Then for \(I:= \sum _{\ell =1}^L J_\ell + J'\) (with the implicit identification of \(J_\ell \) and \(J'\) with the set of its labels in I) we have

(A.8)

To summarize, the constraint determined by the characteristic functions on the right hand side of (A.8) amounts to removing from the standard cumulant expansion all terms which have any clusters internal to one of the sets \(J_\ell \). For instance, thanks to Proposition 5.1, if we consider the expectation of the product of two second order Wick polynomials, we get \({\mathbb E}[\mathop {:}\nolimits \!y_1y_2\!\mathop {:}\nolimits \mathop {:}\nolimits \!y_3y_4\!\mathop {:}\nolimits ]= \kappa ( y_1 , y_3 )\kappa ( y_2 , y_4)+ \kappa ( y_1 , y_4 )\kappa ( y_2 , y_3)+\kappa (y_1,y_2,y_3,y_4)\).

Proposition 5.1 turns out to be a powerful technical tool, used several times in the proofs of Proposition 4.1 and Theorem 4.3.

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Lukkarinen, J., Marcozzi, M. & Nota, A. Summability of Connected Correlation Functions of Coupled Lattice Fields. J Stat Phys 171, 189–206 (2018). https://doi.org/10.1007/s10955-018-2000-6

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