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Polynomial Ensembles and Pólya Frequency Functions

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Abstract

We study several kinds of polynomial ensembles of derivative type which we propose to call Pólya ensembles. These ensembles are defined on the spaces of complex square, complex rectangular, Hermitian, Hermitian antisymmetric and Hermitian anti-self-dual matrices, and they have nice closure properties under the multiplicative convolution for the first class and under the additive convolution for the other classes. The cases of complex square matrices and Hermitian matrices were already studied in former works. One of our goals is to unify and generalize the ideas to the other classes of matrices. Here, we consider convolutions within the same class of Pólya ensembles as well as convolutions with the more general class of polynomial ensembles. Moreover, we derive some general identities for group integrals similar to the Harish–Chandra–Itzykson–Zuber integral, and we relate Pólya ensembles to Pólya frequency functions. For illustration, we give a number of explicit examples for our results.

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References

  1. Adhikari, K., Reddy, N.K., Reddy, T.R., Saha, K.: Determinantal point processes in the plane from products of random matrices. Ann. Inst. H. Poincaré Probab. Stat. 52, 16–46 (2016). arXiv:1308.6817 [math.PR]

    MathSciNet  MATH  Google Scholar 

  2. Akemann, G., Burda, Z.: Universal microscopic correlation functions for products of independent Ginibre matrices. J. Phys. A 45, 465201 (2012). arXiv:1208.0187 [math-ph]

    MathSciNet  MATH  Google Scholar 

  3. Akemann, G., Burda, Z., Kieburg, M., Nagao, T.: Universal microscopic correlation functions for products of truncated unitary matrices. J. Phys. A: Math. Theor. 47, 255202 (2014). arXiv:1310.6395 [math-ph]

    MathSciNet  MATH  Google Scholar 

  4. Akemann, G., Ipsen, J.: Recent exact and asymptotic results for products of independent random matrices, Acta Phys. Polanica B 46, 1747–1784 (2015). arXiv:1502.01667 [math-ph]

    MATH  Google Scholar 

  5. Akemann, G., Ipsen, J., Kieburg, M.: Products of rectangular random matrices: singular values and progressive scattering. Phys. Rev. E 88, 052118 (2013). arXiv:1307.7560 [math-ph]

    Google Scholar 

  6. Akemann, G., Kieburg, M., Wei, L.: Singular value correlation functions for products of Wishart matrices. J. Phys. A 46, 275205 (2013). arXiv:1303.5694 [math-ph]

    MathSciNet  MATH  Google Scholar 

  7. Akemann, G., Strahov, E.: Hole probabilities and overcrowding estimates for products of complex Gaussian matrices. J. Stat. Phys. 151, 987–1003 (2013). arXiv:1211.1576 [math-ph]

    MathSciNet  MATH  Google Scholar 

  8. Andréief, K.A.: Notes sur une relation les intégrales définies des produits des fonctions. Mém. de la Soc. Sci. Bordeaux 2, 1–14 (1883)

    Google Scholar 

  9. Berezin, F.A., Karpelevich, F.I.: Zonal spherical functions and Laplace operators on some symmetric spaces. Doklady Akad. Nauk. SSSR 118, 9–12 (1958)

    MathSciNet  MATH  Google Scholar 

  10. Borodin, A.: Biorthogonal ensembles, Nuclear Phys. B 536, 704–732 (1999). arXiv:math/9804027 [math.CA]

    MathSciNet  MATH  Google Scholar 

  11. Bufetov, A.I.: Unitarily invariant ergodic matrices and free probability. Math. Notes 98, 884–890 (2015)

    MathSciNet  MATH  Google Scholar 

  12. Claeys, T., Kuijlaars, A.B.J., Wang, D.: Correlation kernel for sums and products of random matrices. Random Matrices: Theory Appl. 04, 1550017 (2015). arXiv:1505.00610 [math.PR]

    MathSciNet  MATH  Google Scholar 

  13. Faraut, J.: Infinite Dimensional Harmonic Analysis and Probability. In: Dani, S.G., Graczyk, P. (eds.) Probability Measures on Groups: Recent Directions and Trends. Tata Institute of Fundamental Research, Narosa (2006)

    Google Scholar 

  14. Forrester, P.J.: Eigenvalue statistics for product complex Wishart matrices. J. Phys. A 47, 345202 (2014). arXiv:1401.2572 [math-ph]

    MathSciNet  MATH  Google Scholar 

  15. Forrester, P.J., Wang, D.: Muttalib-Borodin ensembles in random matrix theory — realisations and correlation functions. Electron. J. Probab. 22, 54 (2017). arXiv:1502.07147 [math-ph]

    MathSciNet  MATH  Google Scholar 

  16. Gelfand, I.M., Naĭmark, M., A.: Unitäre Darstellungen der klassischen Gruppen, Akademie-Verlag, Berlin (1957). Translated from Russian: Trudy Mat. Inst. Steklov. 36, 1–288 (1950)

  17. Gross, K.I., Richards, D.S.P.: Total positivity, spherical series, and hypergeometric functions of matrix argument. J. Approx. Theory 59, 224–246 (1989)

    MathSciNet  MATH  Google Scholar 

  18. Guhr, T., Wettig, T.: An Itzykson–Zuber-like integral and diffusion for complex ordinary and supermatrices. J. Math. Phys. 37, 6395–6413 (1996). arXiv:hep-th/9605110

    MathSciNet  MATH  Google Scholar 

  19. Harish-Chandra: Invariant differential operators on a semisimple Lie algebra. Proc. Natl. Acad. Sci. USA 42, 252–253 (1956)

    MathSciNet  MATH  Google Scholar 

  20. Harnad, J., Orlov, A.Y.: Fermionic construction of partition functions for two-matrix models and perturbative Schur function expansions. J. Phys. A 39, 8783–8809 (2006). [arXiv:math-ph/0512056]]

    MathSciNet  MATH  Google Scholar 

  21. Helgason, S.: Groups and Geometric Analysis. Integral geometry, Invariant Differential Operators, and Spherical Functions, Corrected reprint of the 1984 original, Mathematical Surveys and Monographs, vol. 83, American Mathematical Society, Providence, RI (2000)

  22. Hua, L.K.: Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. American Mathematical Society, Providence, RI (1963)

    Google Scholar 

  23. Ipsen, J., Kieburg, M.: Weak commutation relations and eigenvalue statistics for products of rectangular random matrices. Phys. Rev. E 89, 032106 (2014). [arXiv:1310.4154 [math-ph]]

    Google Scholar 

  24. Itzykson, C., Zuber, J.B.: The planar approximation II. J. Math. Phys. 21, 411–21 (1980)

    MathSciNet  MATH  Google Scholar 

  25. Jorgenson, J., Lang, S.: Spherical Inversion on \({\rm SL}_n(\mathbb{R})\), Springer Monographs in Mathematics. Springer, New York (2001)

    Google Scholar 

  26. Kallenberg, O.: Schoenberg’s theorem and unitarily invariant random arrays. J. Theor. Probab. 25, 1013–1039 (2012)

    MathSciNet  MATH  Google Scholar 

  27. Karlin, S.: Polya type distributions, II. Ann. Math. Stat. 28, 281–308 (1957)

    MATH  Google Scholar 

  28. Karlin, S.: Total Positivity, vol. I. Stanford University Press, Stanford, CA (1968)

    MATH  Google Scholar 

  29. Kieburg, M., Kösters, H.: Exact relation between the singular value and eigenvalue statistics. Random Matrices: Theory Appl. 5, 1650015 (2016). arXiv:1601.02586 [math.CA]

    MathSciNet  MATH  Google Scholar 

  30. Kieburg, M., Kösters, H.: Products of random matrices from polynomial ensembles. Ann. Inst. Henri Poincaré Probab. Stat. 55, 98–126 (2019). arXiv:1601.02586 [math.CA]

    MathSciNet  MATH  Google Scholar 

  31. Kieburg, M., Kuijlaars, A.B.J., Stivigny, D.: Singular value statistics of matrix products with truncated unitary matrices. Int. Math. Res. Notices 2016, 3392–3424 (2016). [arXiv:1501.03910 [math.PR]]

    MathSciNet  MATH  Google Scholar 

  32. Kuijlaars, A.B.J.: Transformations of polynomial ensembles. Contemp. Math. 661, 253–268 (2016). [arXiv:1501.05506 [math.PR]]

    MathSciNet  MATH  Google Scholar 

  33. Kuijlaars, A.B.J., Román, P.: Spherical functions approach to sums of random Hermitian matrices. Int. Math. Res. Not. 2019, 1005–1029 (2019). [arXiv:1611.08932 [math.PR]]

    MathSciNet  MATH  Google Scholar 

  34. Kuijlaars, A.B.J., Stivigny, D.: Singular values of products of random matrices and polynomial ensembles. Random Matrices Theory Appl. 3, 1450011 (2014). [arXiv:1404.5802 [math.PR]]

    MathSciNet  MATH  Google Scholar 

  35. Kuijlaars, A.B.J., Zhang, L.: Singular values of products of Ginibre random matrices, multiple orthogonal polynomials and hard edge scaling limits. Commun. Math. Phys. 332, 759–781 (2014). [arXiv:1308.1003 [math-ph]]

    MathSciNet  MATH  Google Scholar 

  36. Liu, D.-Z., Wang, D., Zhang, L.: Bulk and soft-edge universality for singular values of products of Ginibre random matrices. Ann. Inst. H. Poincaré Probab. Stat. 52, 1734–1762 (2014). [arXiv:1412.6777 [math.PR]]

    MathSciNet  MATH  Google Scholar 

  37. Mehta, M.L.: Random Matrices, 3rd edn. Elsevier/Academic Press, Amsterdam (2004)

    MATH  Google Scholar 

  38. Muttalib, K.A.: Random matrix models with additional interactions. J. Phys. A 28, L159–L164 (1995)

    MathSciNet  Google Scholar 

  39. Olshanski, G., Vershik, A.: Ergodic unitarily invariant measures on the space of infinite Hermitian matrices, Contemporary Mathematical Physics (R.L. Dobroshin, R.A. Minlos, M.A. Shubin, A.M. Vershik). Am. Math. Soc. Trans. (2) 175, 137–175 (1996)

    Google Scholar 

  40. Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  41. Orlov, A.Y.: New solvable matrix integrals. Int. J. Mod. Phys. A 19, 276–293 (2004). arXiv:nlin/0209063

    MathSciNet  MATH  Google Scholar 

  42. Pickrell, D.: Mackey analysis of infinite classical motion groups. Pacific J. Math. 150, 139–166 (1991)

    MathSciNet  MATH  Google Scholar 

  43. Pólya, G.: Über Annäherung durch Polynome mit lauter reellen Wurzeln. Rend. di Palermo 36, 279–295 (1913)

    MATH  Google Scholar 

  44. Pólya, G.: Algebraische Untersuchungen über ganze Funktionen vom Geschlechte Null und Eins. J. für Mathematik 145, 224–249 (1915)

    MathSciNet  MATH  Google Scholar 

  45. Schoenberg, I.J.: On Pólya frequency functions I. The totally positive functions and their Laplace transforms. J. d’Analyse Mathématique 1, 331–374 (1951)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We want to thank Gernot Akemann, Friedrich Götze and Arno Kuijlaars for fruitful discussions on this topic. Moreover, we acknowledge support by CRC 701 “Spectral Structures and Topological Methods in Mathematics” as well as by grant AK35/2-1 “Products of Random Matrices”, both funded by Deutsche Forschungsgemeinschaft (DFG). The work of Yanik-Pascal Förster was formerly supported by Studienstiftung des deutschen Volkes.

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Authors and Affiliations

  1. Department of Physics, Bielefeld University, Postfach 100131, D-33501, Bielefeld, Germany

    Yanik-Pascal Förster & Mario Kieburg

  2. Faculty of Natural & Mathematical Sciences, King’s College London, Strand, London, WC2R 2LS, UK

    Yanik-Pascal Förster

  3. School of Mathematics and Statistics, University of Melbourne, 813 Swanton Street, Parkville, Melbourne, VIC 3010, Australia

    Mario Kieburg

  4. Department of Mathematics, Bielefeld University, Postfach 100131, D-33501, Bielefeld, Germany

    Holger Kösters

  5. Institute of Mathematics, University of Rostock, D-18051, Rostock, Germany

    Holger Kösters

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  1. Yanik-Pascal Förster
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Correspondence to Holger Kösters.

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Förster, YP., Kieburg, M. & Kösters, H. Polynomial Ensembles and Pólya Frequency Functions. J Theor Probab 34, 1917–1950 (2021). https://doi.org/10.1007/s10959-020-01030-z

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