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Clustering Bounds on n-Point Correlations for Unbounded Spin Systems

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Abstract

We prove clustering estimates for the truncated correlations, i.e., cumulants of an unbounded spin system on the lattice. We provide a unified treatment, based on cluster expansion techniques, of four different regimes: large mass, small interaction between sites, large self-interaction, as well as the more delicate small self-interaction or near massive Gaussian regime. A clustering estimate in the latter regime is needed for the Bosonic case of the recent result obtained by Lukkarinen and Spohn on the rigorous control on kinetic scales of quantum fluids.

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References

  1. Abdesselam, A.: Feynman diagrams in algebraic combinatorics. Sém. Lothar. Comb. 49 (2002/04), Art. B49c, 45 p. (electronic)

  2. Abdesselam, A., Rivasseau, V.: Trees, forests and jungles: a botanical garden for cluster expansions. In: Constructive Physics, Palaiseau, 1994. Lecture Notes in Physics, vol. 446, pp. 7–36. Springer, Berlin (1995)

    Google Scholar 

  3. Abdesselam, A., Rivasseau, V.: An explicit large versus small field multiscale cluster expansion. Rev. Math. Phys. 9(2), 123–199 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Abdesselam, A., Rivasseau, V.: Explicit fermionic tree expansions. Lett. Math. Phys. 44(1), 77–88 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  5. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Academic Press, New York (2003)

    MATH  Google Scholar 

  6. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and Its Applications, vol. 71. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  7. Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C., Scheffer, G.: Sur les Inégalités de Sobolev Logarithmiques. With a Preface by Dominique Bakry and Michel Ledoux. Panoramas et Synthèses, vol. 10. Soc. Math. France, Paris (2000)

    MATH  Google Scholar 

  8. Bach, V., Møller, J.S.: Correlation at low temperature I. Exponential decay. J. Funct. Anal. 203(1), 93–148 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bach, V., Jecko, T., Sjöstrand, J.: Correlation asymptotics of classical lattice spin systems with nonconvex Hamilton function at low temperature. Ann. H. Poincaré 1(1), 59–100 (2000)

    Article  MATH  Google Scholar 

  10. Balaban, T., Imbrie, J.Z., Jaffe, A.: Effective action and cluster properties of the Abelian Higgs model. Commun. Math. Phys. 114, 257–315 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  11. Balaban, T., Feldman, J., Knörrer, H., Trubowitz, E.: Power series representations for bosonic effective actions. J. Stat. Phys. 134, 839–857 (2009)

    Article  MATH  ADS  Google Scholar 

  12. Battle, G.A., Federbush, P.: A note on cluster expansions, tree graph identities, extra 1/N! factors! Lett. Math. Phys. 8(1), 55–57 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  13. Benfatto, G., Gallavotti, G., Procacci, A., Scoppola, B.: Beta function and Schwinger functions for a many fermions system in one dimension. Anomaly of the Fermi surface. Commun. Math. Phys. 160(1), 93–171 (1994)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. Birnbaum, Z.W.: An inequality for Mill’s ratio. Ann. Math. Stat. 13(2), 245–246 (1942)

    Article  MATH  MathSciNet  Google Scholar 

  15. Brydges, D.C.: A short course on cluster expansions. In: Phénomènes Critiques, Systèmes Aléatoires, Théories de Jauge, Part I, II, Les Houches, 1984, pp. 129–183. North-Holland, Amsterdam (1986)

    Google Scholar 

  16. Brydges, D.C., Federbush, P.: A new form of the Mayer expansion in classical statistical mechanics. J. Math. Phys. 19(10), 2064–2067 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  17. Brydges, D., Kennedy, T.: Mayer expansions and the Hamilton-Jacobi equation. J. Stat. Phys. 48, 19 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  18. Brydges, D., Martin, P.: Coulomb systems at low density: a review. J. Stat. Phys. 96, 1163–1330 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  19. Brydges, D., Dimock, J., Hurd, T.R.: The short distance behavior of (φ 4)3. Commun. Math. Phys. 172, 143–186 (1995)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  20. Caianiello, E.R.: Number of Feynman graphs and convergence. Nuovo Cimento 10(3), 223–225 (1956)

    Google Scholar 

  21. Cammarota, C.: Decay of correlations for infinite range interactions in unbounded spin systems. Commun. Math. Phys. 85(4), 517–528 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  22. Constantinescu, F.: Analyticity in the coupling constant of the λ P(φ) lattice theory. J. Math. Phys. 21(8), 2278–2281 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  23. Duneau, M., Iagolnitzer, D., Souillard, B.: Decrease properties of truncated correlation functions and analyticity properties for classical lattices and continuous systems. Commun. Math. Phys. 31, 191–208 (1973)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  24. Eckmann, J.-P., Magnen, J., Sénéor, R.: Decay properties and Borel summability for the Schwinger functions in P(φ)2 theories. Commun. Math. Phys. 39, 251–271 (1974/75)

    Article  ADS  Google Scholar 

  25. Feldman, J., Magnen, J., Rivasseau, V., Sénéor, R.: A renormalizable field theory: the massive Gross-Neveu model in two dimensions. Commun. Math. Phys. 103(1), 67–103 (1986)

    Article  MATH  ADS  Google Scholar 

  26. Feldman, J., Knörrer, H., Trubowitz, E.: A representation for Fermionic correlation functions. Commun. Math. Phys. 195(2), 465–493 (1998)

    Article  MATH  ADS  Google Scholar 

  27. Fernández, R., Procacci, A.: Cluster expansion for abstract polymer models. New bounds from an old approach. Commun. Math. Phys. 274(1), 123–140 (2007)

    Article  MATH  ADS  Google Scholar 

  28. Gawędzki, K., Kupiainen, A.: Gross-Neveu model through convergent perturbation expansions. Commun. Math. Phys. 102(1), 1–30 (1985)

    Article  ADS  Google Scholar 

  29. Glimm, J., Jaffe, A.: Quantum Physics. A Functional Integral Point of View, 2nd edn. Springer, New York (1987)

    Google Scholar 

  30. Glimm, J., Jaffe, A., Spencer, T.: The particle structure of the weakly coupled P(φ)2 model and other applications of high temperature expansions, part II: The cluster expansion. In: Velo, G., Wightman, A. (eds.) Constructive Quantum Field Theory, Erice, 1973. Lecture Notes in Physics, vol. 25. Springer, New York (1973)

    Google Scholar 

  31. Glimm, J., Jaffe, A., Spencer, T.: The Wightman axioms and particle structure in the P(φ)2 quantum field model. Ann. Math. 100, 585–632 (1974)

    Article  MathSciNet  Google Scholar 

  32. Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics. A Foundation for Computer Science. Advanced Book Program. Addison-Wesley, Reading (1989)

    MATH  Google Scholar 

  33. Gross, L.: Decay of correlations in classical lattice models at high temperature. Commun. Math. Phys. 68(1), 9–27 (1979)

    Article  MATH  ADS  Google Scholar 

  34. Gruber, C., Kunz, H.: General properties of polymer systems. Commun. Math. Phys. 22, 133–161 (1971)

    Article  ADS  MathSciNet  Google Scholar 

  35. Guionnet, A., Zegarlinski, B.: Lectures on logarithmic Sobolev inequalities. In: Séminaire de Probabilités, XXXVI. Lecture Notes in Mathematics, vol. 1801, pp. 1–134. Springer, Berlin (2003)

    Google Scholar 

  36. Helffer, B.: Semiclassical Analysis, Witten Laplacians, and Statistical Mechanics. Series in Partial Differential Equations and Applications, vol. 1. World Scientific, River Edge (2002)

    MATH  Google Scholar 

  37. Helffer, B., Sjöstrand, J.: On the correlation for Kac-like models in the convex case. J. Stat. Phys. 74(1–2), 349–409 (1994)

    Article  MATH  ADS  Google Scholar 

  38. Iagolnitzer, D., Magnen, J.: Asymptotic completeness and multiparticle structure in field theories II. Theories with renormalization: the Gross-Neveu model. Commun. Math. Phys. 111(1), 81–100 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  39. Israel, R.B., Nappi, C.R.: Exponential clustering for long-range integer-spin systems. Commun. Math. Phys. 68(1), 29–37 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  40. Isserlis, L.: On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables. Biometrika 12, 134–139 (1918)

    Google Scholar 

  41. Kunz, H.: Analyticity and clustering properties of unbounded spin systems. Commun. Math. Phys. 59(1), 53–69 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  42. Lesniewski, A.: Effective action for the Yukawa2 quantum field theory. Commun. Math. Phys. 108(3), 437–467 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  43. Lo, A.: On the exponential decay of the n-point correlation functions and the analyticity of the pressure. J. Math. Phys. 48(12), 123506 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  44. Lukkarinen, J., Spohn, H.: Not to normal order—notes on the kinetic limit for weakly interacting quantum fluids. J. Stat. Phys. 134(5–6), 1133–1172 (2009)

    Article  MATH  ADS  Google Scholar 

  45. Lukkarinen, J., Spohn, H.: Weakly nonlinear Schrödinger equation with random initial data. Preprint arXiv:0901.3283v1 [math-ph] (2009)

  46. Mack, G., Pordt, A.: Convergent perturbation expansions for Euclidean quantum field theory. Commun. Math. Phys. 97(1–2), 267–298 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  47. Malyshev, V.A., Minlos, R.A.: Gibbs Random Fields. Cluster Expansions. Translated from the Russian by R. Kotecký and P. Holický. Mathematics and Its Applications (Soviet Series), vol. 44. Kluwer Academic, Dordrecht (1991)

    MATH  Google Scholar 

  48. Mastropietro, V.: Non-perturbative Renormalization. World Scientific, Hackensack (2008)

    MATH  Google Scholar 

  49. Matte, O.: Supersymmetric Dirichlet operators, spectral gaps, and correlations. Ann. H. Poincaré 7(4), 731–780 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  50. Pereira, E., Procacci, A., O’Carroll, M.: Multiscale formalism for correlation functions of fermions. Infrared analysis of the tridimensional Gross-Neveu model. J. Stat. Phys. 95(3–4), 665–692 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  51. Pordt, A.: Mayer expansions for Euclidean lattice field theory: convergence properties and relation with perturbation theory. Desy preprint 85-103, unpublished. Available at http://www-lib.kek.jp/top-e.html (1985)

  52. Procacci, A., Pereira, E.: Infrared analysis of the tridimensional Gross-Neveu model: pointwise bounds for the effective potential. Ann. Inst. H. Poincaré Phys. Théor. 71(2), 129–198 (1999)

    MATH  MathSciNet  Google Scholar 

  53. Procacci, A., Scoppola, B.: On decay of correlations for unbounded spin systems with arbitrary boundary conditions. J. Stat. Phys. 105(3–4), 453–482 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  54. Procacci, A., de Lima, B.N.B., Scoppola, B.: A remark on high temperature polymer expansion for lattice systems with infinite range pair interactions. Lett. Math. Phys. 45(4), 303–322 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  55. Rivasseau, V.: From Perturbative to Constructive Renormalization. Princeton Series in Physics. Princeton University Press, Princeton (1991)

    Google Scholar 

  56. Ruelle, D.: Cluster property of the correlation functions of classical gases. Rev. Mod. Phys. 36, 580–584 (1964)

    Article  ADS  MathSciNet  Google Scholar 

  57. Ruelle, D.: Statistical Mechanics: Rigorous Results. Benjamin, New York/Amsterdam (1969)

    MATH  Google Scholar 

  58. Salmhofer, M.: Renormalization. An Introduction. Texts and Monographs in Physics. Springer, Berlin (1999)

    MATH  Google Scholar 

  59. Salmhofer, M.: Clustering of fermionic truncated expectation values via functional integration. J. Stat. Phys. 134(5–6), 941–952 (2009)

    Article  MATH  ADS  Google Scholar 

  60. Simon, B.: The Statistical Mechanics of Lattice Gases, vol. I. Princeton Series in Physics. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  61. Sjöstrand, J.: Correlation asymptotics and Witten Laplacians. Algebra Anal. 8(1), 160–191 (1996) (translation in St. Petersburg Math. J. 8(1), 123–147 (1997))

    MATH  Google Scholar 

  62. Sjöstrand, J.: Complete asymptotics for correlations of Laplace integrals in the semi-classical limit. Mém. Soc. Math. France (N.S.) 83 (2000)

  63. Sokal, A.: Mean-field bounds and correlation inequalities. J. Stat. Phys. 28(3), 431–439 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  64. Sokal, A.D.: Bounds on the complex zeros of (di)chromatic polynomials and Potts-model partition functions. Comb. Probab. Comput. 10(1), 41–77 (2001)

    MATH  MathSciNet  Google Scholar 

  65. Stanley, R.P.: Enumerative Combinatorics, vol. 2. With a foreword by Gian-Carlo Rota and Appendix 1 by Sergey Fomin. Cambridge Studies in Advanced Mathematics, vol. 62. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  66. Wagner, W.: Analyticity and Borel-summability of the perturbation expansion for correlation functions of continuous spin systems. Helv. Phys. Acta 54(3), 341–363 (1981/1982)

    MathSciNet  Google Scholar 

  67. Wick, G.C.: The evaluation of the collision matrix. Phys. Rev. (2) 80, 268–272 (1950)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  68. Yoshida, N.: The log-Sobolev inequality for weakly coupled lattice fields. Probab. Theory Relat. Fields 115(1), 1–40 (1999)

    Article  MATH  Google Scholar 

  69. Yoshida, N.: The equivalence of the log-Sobolev inequality and a mixing condition for unbounded spin systems on the lattice. Ann. Inst. H. Poincaré Probab. Stat. 37(2), 223–243 (2001)

    Article  MATH  ADS  Google Scholar 

  70. Zegarlinski, B.: The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice. Commun. Math. Phys. 175(2), 401–432 (1996)

    Article  MATH  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Virginia, P.O. Box 400137, Charlottesville, VA, 22904-4137, USA

    Abdelmalek Abdesselam

  2. Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, Caixa Postal 702, 30161-970, Belo Horizonte, MG, Brasil

    Aldo Procacci

  3. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133, Rome, Italy

    Benedetto Scoppola

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  1. Abdelmalek Abdesselam
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  2. Aldo Procacci
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Abdesselam, A., Procacci, A. & Scoppola, B. Clustering Bounds on n-Point Correlations for Unbounded Spin Systems. J Stat Phys 136, 405–452 (2009). https://doi.org/10.1007/s10955-009-9789-y

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