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Universal shocks in the Wishart random-matrix ensemble

Jean-Paul Blaizot1,*, Maciej A. Nowak2,†, and Piotr Warchoł1,3,‡

  • 1Institut de Physique Théorique (IPhT), CNRS/URA 2306, CEA-Saclay, 91191 Gif-sur Yvette, France
  • 2M. Smoluchowski Institute of Physics and Mark Kac Center for Complex Systems Research, Jagiellonian University, PL-30059 Cracow, Poland
  • 3M. Smoluchowski Institute of Physics, Jagiellonian University, PL-30059 Cracow, Poland

  • *Jean-Paul.Blaizot@cea.fr
  • †nowak@th.if.uj.edu.pl
  • ‡piotr.warchol@uj.edu.pl
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Phys. Rev. E 87, 052134 – Published 28 May, 2013

DOI: https://doi.org/10.1103/PhysRevE.87.052134

Abstract

We show that the derivative of the logarithm of the average characteristic polynomial of a diffusing Wishart matrix obeys an exact partial differential equation valid for an arbitrary value of N, the size of the matrix. In the large N limit, this equation generalizes the simple inviscid Burgers equation that has been obtained earlier for Hermitian or unitary matrices. The solution, through the method of characteristics, presents singularities that we relate to the precursors of shock formation in the Burgers equation. The finite N effects appear as a viscosity term in the Burgers equation. Using a scaling analysis of the complete equation for the characteristic polynomial, in the vicinity of the shocks, we recover in a simple way the universal Bessel oscillations (so-called hard-edge singularities) familiar in random-matrix theory.

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See Also

Universal shocks in the Wishart random-matrix ensemble. II. Nontrivial initial conditions

Jean-Paul Blaizot, Maciej A. Nowak, and Piotr Warchoł
Phys. Rev. E 89, 042130 (2014)

Article Text

References (37)

  1. F. J. Dyson, J. Math. Phys. 3, 119 (1962).
  2. J.-P. Blaizot and M. A. Nowak, Phys. Rev. Lett. 101, 102001 (2008).
  3. J.-P. Blaizot and M. A. Nowak, Phys. Rev. E 82, 051115 (2010).
  4. H. Neuberger, Phys. Lett. B 666, 106 (2008).
  5. H. Neuberger, Phys. Lett. B 670, 235 (2008).
  6. S. J. Chapman, C. J. Howls, J. R. King, and A. B. Olde Daalhuis, Nonlinearity 20, 2425 (2007).
  7. R. A. Janik and W. Wieczorek, J. Phys. A: Math. Gen. 37, 6521 (2004).
  8. This universal dynamics has been observed, for example, in the behavior of Wilson loops in simulations of Yang-Mills theory with a large number of colors [37]. In this case, the role of the time is played by the area of the Wilson loop and the size of the matrix is determined by the number of colors. The critical oscillations of the preshock are associated to the onset of nonperturbative effects of the Wilson loop, i.e., they signal the transition to the strong coupling regime.
  9. J. Wishart, Biometrika 20, 32 (1928).
  10. Zh. Bai and J. W. Silverstein, Spectral Analysis of Large Dimensional Random Matrices (Science Press, Beijing, 2011).
  11. S. S. Wilks, Mathematical Statistics (Wiley, New York, London, 1963); R. W. Preisendorfer, Principal Component Analysis in Meteorology and Oceanography (Elsevier, Amsterdam, 1988); J.-P. Bouchaud and M. Potters, Theory of Financial Risks (Cambridge University Press, Cambridge, UK, 2001); N. Patterson, A. L. Price, and D. Reich, PLOS Genetics 2, 2074 (2006).
  12. J. J. M. Verbaarschot, The Oxford Handbook of Random Matrix Theory, edited by G. Akermann, J. Baik, and P. Di Francesco (Oxford University Press, Oxford, 2011).
  13. G. J. Foschini, Bell Labs Tech. J. 1, 42 (1996).
  14. E. Telatar, Euro. Trans. Telecom. 10, 585 (1999).
  15. R. Couillet and M. Debbah, Random Matrix Methods for Wireless Communications (Cambridge University Press, Cambridge, UK, 2011).
  16. I. Bengtsson and K. Życzkowski, Geometry of Quantum States (Cambridge University Press, Cambridge, UK, 2006).
  17. C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997), and references therein.
  18. P. W. Brouwer, K. M. Frahm, and C. W. J. Beenakker, Phys. Rev. Lett. 78, 4737 (1997).
  19. B. V. Bronk, J. Math. Phys. 6, 228 (1965).
  20. V. A. Marcenko and L. A. Pastur, Math. USSR-Sb. 1, 457 (1967).
  21. D. V. Voiculescu, K. J. Dykema, and A. Nica, Free Random Variables, CRM Monograph Series, Vol. 1 (American Mathematical Society, Providence, 1992).
  22. W. König and N. O’Connell, Electron Commun. Prob. 6, 107 (2001).
  23. M. F. Bru, J. Multivariate Anal. 29, 127 (1989).
  24. M. F. Bru, J. Theor. Prob. 4, 725 (1991).
  25. T. Akuzawa and M. Wadati, Chaos Solitons Fractals 8, 99 (1997).
  26. M. Katori and H. Tanemura, J. Math. Phys. 45, 3058 (2004), and references therein.
  27. C. Nadal and S. N. Majumdar, Phys. Rev. E 79, 061117 (2009).
  28. R. Allez, J.-Ph. Bouchaud, S. N. Majumdar, and P. Vivo, J. Phys. A: Math. Theor. 46, 015001 (2013).
  29. T. Cabanal Duvillard and A. Guionnet, Ann. Prob. 29, 1205 (2001).
  30. J. Feinberg and A. Zee, J. Stat. Phys. 87, 473 (1997).
  31. See, e.g., F. Hiai and D. Petz, The Semicircle Law, Free Random Variables and Entropy (American Mathematical Society, Providence, 2000).
  32. E. Brezin and S. Hikami, Comm. Math. Phys. 214, 111 (2000).
  33. Y. V. Fyodorov and E. Strahov, J. Phys. A: Math. Gen. 36, 3203 (2003).
  34. J.-P. Blaizot, M. A. Nowak, and P. Warchoł (unpublished).
  35. G. Akemann and Y. V. Fyodorov, Nucl. Phys. B 664, 457 (2003).
  36. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1964).
  37. R. Narayanan and H. Neuberger, J. High Energy Phys. 12 (2007) 066.

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Phys. Rev. E 87, 052134– Published 28 May, 2013
  • Received 23 November 2012

©2013 American Physical Society

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