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Mod-Gaussian Convergence and Its Applications for Models of Statistical Mechanics

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Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 2137)

Abstract

In this paper we complete our understanding of the role played by the limiting (or residue) function in the context of mod-Gaussian convergence. The question about the probabilistic interpretation of such functions was initially raised by Marc Yor. After recalling our recent result which interprets the limiting function as a measure of “breaking of symmetry” in the Gaussian approximation in the framework of general central limit theorems type results, we introduce the framework of L1-mod-Gaussian convergence in which the residue function is obtained as (up to a normalizing factor) the probability density of some sequences of random variables converging in law after a change of probability measure. In particular we recover some celebrated results due to Ellis and Newman on the convergence in law of dependent random variables arising in statistical mechanics. We complete our results by giving an alternative approach to the Stein method to obtain the rate of convergence in the Ellis-Newman convergence theorem and by proving a new local limit theorem. More generally we illustrate our results with simple models from statistical mechanics.

Keywords

  • Limit Theorem
  • Ising Model
  • Moment Generate Function
  • Transfer Matrix Method
  • Dependent Random Variable

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In memoriam, Marc Yor

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References

  1. R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic, London, 1982)

    MATH  Google Scholar 

  2. A. Bovier, Statistical Mechanics of Disordered Systems. Cambridge Series in Statistical and Probabilistic Mechanics, vol. 18 (Cambridge University Press, Cambridge, 2006)

    Google Scholar 

  3. F. Delbaen, E. Kowalski, A. Nikeghbali, Mod-ϕ convergence (2011) (arXiv:1107.5657v2 [math.PR])

    Google Scholar 

  4. P. Eichelsbacher, M. Löwe, Stein’s method for dependent random variables occurring in statistical mechanics. Electron. J. Probab. 15(30), 962–988 (2010)

    MATH  MathSciNet  Google Scholar 

  5. R.S. Ellis, C.M. Newman, Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrsch. Verw. Gebiete 44(2), 117–139 (1978)

    MATH  MathSciNet  CrossRef  Google Scholar 

  6. R.S. Ellis, C.M. Newman, J.S. Rosen, Limit theorems for sums of dependent random variables occurring in statistical mechanics II. Z. Wahrsch. Verw. Gebiete 51(2), 153–169 (1980)

    MATH  MathSciNet  CrossRef  Google Scholar 

  7. C.-G. Esseen, Fourier analysis of distribution functions. A mathematical study of the Laplace-Gauss law. Acta Math. 77(1), 1–125 (1945)

    MATH  MathSciNet  Google Scholar 

  8. V. Féray, P.-L. Méliot, A. Nikeghbali, Mod-ϕ convergence and precise deviations (2013) (arXiv:1304.2934v3 [math.PR])

    Google Scholar 

  9. I.A. Ibragimov, Y.V. Linnik, Independent and Stationary Sequences of Random Variables (Wolters-Noordhoff, Groningen, 1971)

    MATH  Google Scholar 

  10. J. Jacod, E. Kowalski, A. Nikeghbali, Mod-Gaussian convergence: new limit theorems in probability and number theory. Forum Math. 23(4), 835–873 (2011)

    MATH  MathSciNet  CrossRef  Google Scholar 

  11. D. Knuth, The Art of Computer Programming. Generating All Trees - History of Combinatorial Generation, vol. 4 [Fascicle 4] (Addison Wesley Longman, Amsterdam, 2004)

    Google Scholar 

  12. E. Kowalski, A. Nikeghbali, Mod-Poisson convergence in probability and number theory. Int. Math. Res. Not. 18, 3549–3587 (2010)

    MathSciNet  Google Scholar 

  13. E. Kowalski, A. Nikeghbali, Mod-Gaussian distribution and the value distribution of ζ(1∕2 + it) and related quantities. J. Lond. Math. Soc. 86(2), 291–319 (2012)

    MATH  MathSciNet  CrossRef  Google Scholar 

  14. E. Kowalski, J. Najnudel, A. Nikeghbali, A characterization of limiting functions arising in mod-∗ convergence (2013). http://arxiv.org/pdf/1304.2179.pdf

  15. V.P. Leonov, A.N. Shiryaev, On a method of calculation of semi-invariants. Theory Probab. Appl. 4, 319–329 (1959)

    MATH  CrossRef  Google Scholar 

  16. E. Lukacs, O. Szász, On analytic characteristic functions. Pac. J. Math. 2(4), 615–625 (1952)

    MATH  CrossRef  Google Scholar 

  17. M. Reed, B. Simon, Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-adjointness (Academic, New York, 1975)

    Google Scholar 

  18. A. Röllin, N. Ross, Local limit theorems via Landau-Kolmogorov inequalities. Bernoulli 21(2), 851–880 (2015) (arXiv:1011.3100v2 [math.PR])

    Google Scholar 

  19. G.-C. Rota, On the foundations of combinatorial theory I: theory of Möbius functions. Zeit. Wahr. Verw. Geb. 2, 340–368 (1964)

    MATH  MathSciNet  CrossRef  Google Scholar 

  20. D. Tamari, The algebra of bracketings and their enumeration. Nieuw Archief voor Wiskunde Ser. 3 10, 131–146 (1962)

    MathSciNet  Google Scholar 

  21. T. Tao, Topics in Random Matrix Theory. Graduate Studies in Mathematics, vol 132 (American Mathematical Society, Providence, 2012)

    Google Scholar 

  22. V.A. Zorich, Mathematical Analysis II. Universitext (Springer, New York, 2004)

    MATH  Google Scholar 

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Acknowledgements

The authors would like to thank M. Carston and C. Newman for fruitful discussions on the models coming from statistical mechanics; and P.-O. Dehaye and V. Féray for comments on the combinatorics of the cumulants of the one-dimensional Ising model. We would also like to thank the anonymous referee for his valuable comments, that have allowed us in particular to correct a false statement of Theorem 13 that appeared in an earlier version of this paper.

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Correspondence to Ashkan Nikeghbali .

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Méliot, PL., Nikeghbali, A. (2015). Mod-Gaussian Convergence and Its Applications for Models of Statistical Mechanics. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) In Memoriam Marc Yor - Séminaire de Probabilités XLVII. Lecture Notes in Mathematics(), vol 2137. Springer, Cham. https://doi.org/10.1007/978-3-319-18585-9_17

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