# Summability of Connected Correlation Functions of Coupled Lattice Fields

- 70 Downloads

## 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.

## References

- 1.Abdesselam, A., Procacci, A., Scoppola, B.: Clustering bounds on \(n\)-point correlations for unbounded spin systems. J. Stat. Phys.
**136**(3), 405–452 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar - 2.Abrahamsen, P.: A review of Gaussian random fields and correlation functions. Technical Report 917, Norwegian Computing Center, Oslo (1997)Google Scholar
- 3.Abramowitz, M., Stegun, I.A.: Handbook of Mathematical functions, Dover edn. Dover Publications, New York (1972)MATHGoogle Scholar
- 4.Adler, R.J., Taylor, J.E.: Random Fields and Geometry. Springer, New York (2007)MATHGoogle Scholar
- 5.Amdeberhan, T., Moll, V.H., Vignat, C.: A probabilistic interpretation of a sequence related to Narayana numbers. J. Anal. Comb. (2013)Google Scholar
- 6.Bodineau, T., Gallagher, I., Saint-Raymond, L.: From hard sphere dynamics to the Stokes-Fourier equations: an \(L^2\) analysis of the Boltzmann-Grad limit. Ann. PDE.
**1**(3), 2–119 (2017)CrossRefGoogle Scholar - 7.Bodineau, T., Gallagher, I., Saint-Raymond, L.: The Brownian motion as the limit of a deterministic system of hard-spheres. Invent. Math.
**203**(2), 493–553 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar - 8.Bogachev, V.I.: Gaussian Measures. American Mathematical Society, New York (1998)CrossRefMATHGoogle Scholar
- 9.Chen, T.: Localization lengths and Boltzmann limit for the Anderson model at small disorders in dimension 3. J. Stat. Phys.
**120**, 279–337 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar - 10.Erdős, L., Salmhofer, M., Yau, H.-T.: Quantum diffusion of the random Schrödinger evolution in the scaling limit I. The non-recollision diagrams. Acta Math.
**200**(2), 211–277 (2008)MathSciNetCrossRefMATHGoogle Scholar - 11.Erdős, L., Yau, H.-T.: Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation. Commun. Pure Appl. Math.
**53**(6), 667–735 (2000)CrossRefMATHGoogle Scholar - 12.Glimm, J., Jaffe, A.: Quantum Physics: A Functional Integral Point of View, Second edn. Springer, New York (1987)CrossRefMATHGoogle Scholar
- 13.Janson, S.: Gaussian Hilbert Spaces. Cambridge University Press, Cambridge (1997)CrossRefMATHGoogle Scholar
- 14.Lebowitz, J.L., Penrose, O.: Analytic and clustering properties of thermodynamic functions and distribution functions for classical lattice and continuum systems. Commun. Math. Phys.
**11**(2), 99–124 (1968)ADSMathSciNetCrossRefMATHGoogle Scholar - 15.Lévy, P.: Processus Stochastiques et Mouvement Brownien. Suivi d’une note de M. Loève. Gauthier-Villars, Paris (1948)MATHGoogle Scholar
- 16.Loève, M.: Probability Theory. II. Graduate Texts in Mathematics, vol. 46, fourth edn. Springer, New York (1978)CrossRefMATHGoogle Scholar
- 17.Lukkarinen, J.: Kinetic theory of phonons in weakly anharmonic particle chains. In: Lepri, S. (ed.) Thermal Transport in Low Dimensions: from Statistical Physics to Nanoscale Heat Transfer. Lecture Notes in Physics, vol. 921, pp. 159–214. Springer, New York (2016)Google Scholar
- 18.Lukkarinen, J., Marcozzi, M.: Wick polynomials and time-evolution of cumulants. J. Math. Phys.
**57**, 083301 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar - 19.Lukkarinen, J., Spohn, H.: Weakly nonlinear Schrödinger equation with random initial data. Invent. Math.
**183**(1), 79–188 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar - 20.Lukkarinen, J., Spohn, H.: Kinetic limit for wave propagation in a random medium. Arch. Ration. Mech. Anal.
**183**(1), 93–162 (2007)MathSciNetCrossRefMATHGoogle Scholar - 21.Peccati, G., Taqqu, M.S.: Wiener Chaos: Moments, Cumulants and Diagrams: A Survey with Computer Implementation. Springer, New York (2011)CrossRefMATHGoogle Scholar
- 22.Pulvirenti, M., Simonella, S.: The Boltzmann-Grad limit of a hard sphere system: analysis of the correlation error. Invent. Math.
**207**(3), 1135–1237 (2017)ADSMathSciNetCrossRefMATHGoogle Scholar - 23.Ruelle, D.: Statistical Mechanics: Rigorous Results. W. A. Benjamin Inc, New York (1969)MATHGoogle Scholar
- 24.Simon, B.: The statistical mechanics of lattice gases. Princeton Series in Physics, vol. I. Princeton University Press, Princeton, NJ (1993)MATHGoogle Scholar
- 25.Spohn, H.: The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics. J. Stat. Phys.
**124**(2–4), 1041–1104 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar - 26.Spohn, H.: Large Scale Dynamics of Interacting Particles. Springer, Berlin (1991)CrossRefMATHGoogle Scholar
- 27.Villani, C.: Optimal Transport, Old and New. Grundlehren der Mathematischen Wissenschaften. Springer, New York (2009)CrossRefMATHGoogle Scholar