Advertisement

Probability Theory and Related Fields

, Volume 175, Issue 3–4, pp 833–847 | Cite as

A generalisation of the relation between zeros of the complex Kac polynomial and eigenvalues of truncated unitary matrices

  • Peter J. ForresterEmail author
  • Jesper R. Ipsen
Article
  • 61 Downloads

Abstract

The zeros of the random Laurent series \(1/\mu - \sum _{j=1}^\infty c_j/z^j\), where each \(c_j\) is an independent standard complex Gaussian, is known to correspond to the scaled eigenvalues of a particular additive rank 1 perturbation of a standard complex Gaussian matrix. For the corresponding random Maclaurin series obtained by the replacement \(z \mapsto 1/z\), we show that these same zeros correspond to the scaled eigenvalues of a particular multiplicative rank 1 perturbation of a random unitary matrix. Since the correlation functions of the latter are known, by taking an appropriate limit the correlation functions for the random Maclaurin series can be determined. Only for \(|\mu | \rightarrow \infty \) is a determinantal point process obtained. For the one and two point correlations, by regarding the Maclaurin series as the limit of a random polynomial, a direct calculation can also be given.

Mathematics Subject Classification

26C10 15B52 30B20 

Notes

Acknowledgements

The work is part of a research program supported by the Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers. PJF also acknowledges partial support from the Australian Research Council Grant DP170102028.

References

  1. 1.
    Adler, R.J., Taylor, J.E.: Random Fields and Geometry. Springer, New York (2007)zbMATHGoogle Scholar
  2. 2.
    Butez, R.: The largest root of random Kac polynomials is heavy tailed. Electron. Commun. Probab. 23 (2018), paper no. 20Google Scholar
  3. 3.
    Diaconis, P., Forrester, P.J.: Hurwitz and the origin of random matrix theory in mathematics. Random Matrix Theory Appl. 6, 1730001 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Forrester, P.J.: Log-Gases and Random Matrices. Princeton University Press, Princeton (2010)CrossRefGoogle Scholar
  5. 5.
    Forrester, P.J.: The limiting Kac random polynomial and truncated random orthogonal matrices. J. Stat. Mech. 2010, P12018 (2010)CrossRefGoogle Scholar
  6. 6.
    Fyodorov, Y.V.: Spectra of random matrices close to unitary and scattering theory for discrete-time systems. Disordered and complex systems. In: AIP Conference Proceedings, vol. 553, pp. 191–196. American Institute of Physics, Melville (2001)Google Scholar
  7. 7.
    Fyodorov, Y.V., Sommers, H.-J.: Random matrices close to hermitian or unitary: overview of methods and results. J. Phys. A 36, 3303–3347 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hannay, J.H.: Chaotic analytic zero points: exact statistics for those of a random spin state. J. Phys. A 29, L101–L105 (1996)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Zeros of Gaussian Analytic Functions and Determinantal Point Processes. American Mathematical Society, Providence (2009)CrossRefGoogle Scholar
  10. 10.
    Ishikawa, M., Kawamuko, H., Okanda, S.: A Paffian–Hafnian analogue of Borchardt’s identity. Electron. J. Comb. 12, 9 (2005)zbMATHGoogle Scholar
  11. 11.
    Kac, M.: On the average number of real roots of a random algebraic equation. Bull. Am. Math. Soc. 49, 314 (1943)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Krishnapur, M.: Zeros of random analytic functions. Ann. Prob. 37, 314–346 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Matsumoto, S., Shirai, T.: Correlation functions for zeros of Gaussian power series and Pfaffians. Electron. J. Probab. 18 (2013), paper no. 49Google Scholar
  14. 14.
    Peres, Y., Virag, B.: Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process. Acta. Math. 194, 1–35 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Poplavskyi, M., Schehr, G.: Exact persistence exponent for the \(2d\)-diffusion equation and related Kac polynomials. Phys. Rev. Lett. 121, 150601 (2018)CrossRefGoogle Scholar
  16. 16.
    Prosen, T.: Exact statistics of complex zeros for Gaussian random polynomials with real coefficients. J. Phys. A 29, 4417–4423 (1996)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Rajan, K., Abbott, L.F.: Eigenvalue spectra of random matrices for neural networks. Phys. Rev. Lett. 97, 188104 (2006)CrossRefGoogle Scholar
  18. 18.
    Singer, D.: A bijective proof of Borchardt’s identity. Electr. J. Comb. 11, 48 (2004)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Tao, T.: Outliers in the spectrum of iid matrices with bounded rank perturbations. Probab. Theory Relat. Fields 155, 231–263 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Wei, Y., Fyodorov, Y.V.: On the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values. J. Phys. A 41, 502001 (2008)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Zyczkowski, K., Sommers, H.-J.: Truncations of random unitary matrices. J. Phys. A 33, 2045–2057 (2000)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and Statistics, ARC Centre of Excellence for Mathematical and Statistical FrontiersThe University of MelbourneMelbourneAustralia

Personalised recommendations