Vol. 18 No. 11 (2026): On Mixing Properties of the Limiting Gibbs Process

Published: 2026-09-23

Articles

  • Articles

    On Mixing Properties of the Limiting Gibbs Process

    Suren Poghosyan
    View PDF
    Abstract

    We consider locally compact second countable Abelian topological group E with Haar measure and pair potentials in E satisfying some natural stability and regularity conditions. We show that the limiting Gibbs process $\operatorname{G}$ with empty boundary conditions is strongly Brillinger-mixing in the spirit of Lothar Heinrich and thereby mixing with respect to the group of translations and ergodic.

    References

    D.R. Brillinger, Statistical inference for stationary point processes. In: M.L. Puri (ed.), Stochastic Processes and Related Topics, Academic Press, pp. 55-99, 1975.

    H.-O. Georgii, Canonical and grand canonical Gibbs states for continuum systems. Commun. Math. Phys., 48 (1976), pp. 31-51.

    L. Heinrich, On the strong Brillinger-mixing property of α-determinantal point processes and some applications. Appl. Math., 61 (2016), pp. 443-461.

    L. Heinrich, Brillinger-mixing point processes need not to be ergodic. Stat. Probab. Lett., 138 (2018), pp. 31-35.

    G. Ivanoff, Central limit theorems for point processes. Stoch. Proc. Appl., 12 (1982), no. 2, pp. 171-186.

    K. Krickeberg, Moments of point processes. In: E.F. Harding and D.G. Kendall (ed.), Stochastic Geometry, Wiley, pp. 243-313, 1974.

    Y.G. Kondratiev, R.A. Minlos, M. Rockner and G.V. Shchepan'uk, Exponential mixing for classical continuous systems. Preprint of Diskrete Strukturen in der Mathematik SFB 343, Universitat Bielefeld, pp. 243-254 (1999).

    K. Matthes, J. Kerstan and J. Mecke, Infinitely Divisible Point Processes. Wiley, 1978.

    J. Mecke, Stationäre zufällige Maße auf lokalkompakten Abelschen gruppen. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 9 (1967), pp. 36-58.

    J. Mecke, Random Measures. Classical Lectures, Walter Warmuth Verlag, 2011.

    R.A. Minlos, Gibbs limit distribution. Funct. Anal. Appl., 1 (1967), pp. 141-150.

    R.A. Minlos, Regularity of the Gibbs limit distribution. Funct. Anal. Appl., 1 (1967), no. 3, pp. 206-217.

    B. Nahapetian, Limit Theorems and Some Applications in Statistical Physics, Teubner Texte zur Mathematik Band 123, 1991.

    B. Nehring, Construction of classical and quantum Gases. The method of cluster expansions. Mathematical lessons, Walter Warmuth Verlag, 2013.

    X.X. Nguyen and H. Zessin, Integral and differential characterizations of the Gibbs process. Math. Nachr., 88 (1979), pp. 105-115.

    S. Poghosyan and D. Ueltschi, Abstract cluster expansion with applications to statistical mechanical systems. J. Math. Phys., 50 (2009), no. 5, article 053509.

    S. Poghosyan and H. Zessin, Cluster representation of classical and quantum processes. Moscow Math. J., 19 (2019), pp. 1-19.

    S. Poghosyan and H. Zessin, Penrose-stable interactions in classical statistical mechanics. Ann. Henri Poincaré, 23 (2022), pp. 739-771.

    A. Procacci and S. Yuhjtman, Convergence of Mayer and Virial expansions and the Penrose tree-graph identity. Lett. Math. Phys., 107 (2017), no. 1, pp. 31-46.

    P. Ressel and W. Schmidtchen,A new characterization of Laplace functionals and probability generating functionals. Probab. Theory Relat. Fields, 88 (1991), pp. 195-213.

    D. Ruelle, Statistical Mechanics, Rigorous Results, 3rd edition, Imperial College Press and World Scientific Publishing, 1999.

    H. Spohn, Equilibrium fluctuations for interacting Brownian particles. Commun. Math. Phys., 103 (1986), pp. 1-33.

    Y. Takahashi, Random point fields revisited: Fock space associated with Poisson measures, fermion (determinantal) processes and Gibbs measures. Research Institute for Mathematical Sciences, Kyoto University, 1681 (2009).

    D. Ueltschi, An improved tree-graph bound, preprint arXiv:1705.05353, 2017.

    M. Westcott, The probability generating functional. J. Australian Math. Soc., 14 (1972), pp. 448-466.