The spectral edge of unitary Brownian motion

Article

Abstract

The Brownian motion \((U^N_t)_{t\ge 0}\) on the unitary group converges, as a process, to the free unitary Brownian motion \((u_t)_{t\ge 0}\) as \(N\rightarrow \infty \). In this paper, we prove that it converges strongly as a process: not only in distribution but also in operator norm. In particular, for a fixed time \(t>0\), we prove that the unitary Brownian motion has a spectral edge: there are no outlier eigenvalues in the limit. We also prove an extension theorem: any strongly convergent collection of random matrix ensembles independent from a unitary Brownian motion also converge strongly jointly with the Brownian motion. We give an application of this strong convergence to the Jacobi process.

Mathematics Subject Classification

15B52 46L54 60F15 60G55 60J65 

References

  1. 1.
    Anderson, G.W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices, vol. 118. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2010)Google Scholar
  2. 2.
    Bai, Z. D.: Methodologies in spectral analysis of large-dimensional random matrices, a review. Stat. Sin. 9(3), 611–677 (1999). With comments by G. J. Rodgers and Jack W. Silverstein; and a rejoinder by the authorGoogle Scholar
  3. 3.
    Bai, Z.D., Yin, Y.Q.: Necessary and sufficient conditions for almost sure convergence of the largest eigenvalue of a Wigner matrix. Ann. Probab. 16(4), 1729–1741 (1988)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Biane, P.: Free Brownian motion, free stochastic calculus and random matrices. Free Probability Theory (Waterloo, ON, 1995). Fields Institute Communications, vol. 12, pp. 1–19. American Mathematical Society, Providence (1997)Google Scholar
  5. 5.
    Biane, P.: Segal-Bargmann transform, functional calculus on matrix spaces and the theory of semi-circular and circular systems. J. Funct. Anal. 144(1), 232–286 (1997)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Biane, P., Speicher, R.: Stochastic calculus with respect to free Brownian motion and analysis on Wigner space. Probab. Theory Relat. Fields 112(3), 373–409 (1998)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Biane, P., Speicher, R.: Free diffusions, free entropy and free Fisher information. Ann. Inst. H. Poincaré Probab. Stat. 37(5), 581–606 (2001)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cébron, G.: Free convolution operators and free Hall transform. J. Funct. Anal. 265(11), 2645–2708 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cébron, G., Kemp, T.: Fluctuations of Brownian motions on \({\mathbb{G}L}_N\) (2014). arXiv:1409.5624
  10. 10.
    Collins, B.: Moments and cumulants of polynomial random variables on unitary groups, the Itzykson–Zuber integral, and free probability. Int. Math. Res. Not. 17, 953–982 (2003)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Collins, B.: Product of random projections, Jacobi ensembles and universality problems arising from free probability. Probab. Theory Relat. Fields 133(3), 315–344 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Collins, B., Kemp, T.: Liberation of projections. J. Funct. Anal. 266(4), 1988–2052 (2014)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Collins, B., Male, C.: The strong asymptotic freeness of Haar and deterministic matrices. Ann. Sci. Éc. Norm. Supér. (4) 47 1, 147–163 (2014)Google Scholar
  14. 14.
    Dahlqvist, A.: Integration formula for Brownian motion on classical compact Lie groups. Ann. Inst. H. Poincaré Probab. Stat. (to appear)Google Scholar
  15. 15.
    Demni, N.: Free Jacobi process. J. Theor. Probab. 21(1), 118–143 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Demni, N.: Free Jacobi process associated with one projection: local inverse of the flow (2015). arXiv:1502.00013
  17. 17.
    Demni, N., Hamdi, T., Hmidi, T.: Spectral distribution of the free Jacobi process. Indiana Univ. Math. J. 61(3), 1351–1368 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Demni, N., Hmidi, T.: Spectral distribution of the free Jacobi process associated with one projection. Colloq. Math. 137(2), 271–296 (2014)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Demni, N., Zani, M.: Large deviations for statistics of the Jacobi process. Stoch. Process. Appl. 119(2), 518–533 (2009)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Driver, B.K., Hall, B.C., Kemp, T.: The large-\(N\) limit of the Segal–Bargmann transform on \(\mathbb{U}_N\). J. Funct. Anal. 265(11), 2585–2644 (2013)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Erdős, L., Farrell, B.: Local eigenvalue density for general MANOVA matrices. J. Stat. Phys. 152(6), 1003–1032 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Evans, L.C.: Partial Differential Equations, vol. 19, 2nd edn. Graduate Studies in Mathematics. American Mathematical Society, Providence (2010)Google Scholar
  23. 23.
    Gustavsson, J.: Gaussian fluctuations of eigenvalues in the GUE. Ann. Inst. H. Poincaré Probab. Stat. 41(2), 151–178 (2005)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Haagerup, U., Thorbjørnsen, S.: A new application of random matrices: \({\rm Ext}(C^*_{\rm red}(F_2))\) is not a group. Ann. Math. (2) 162(2), 711–775 (2005)Google Scholar
  25. 25.
    Hiai, F., Miyamoto, T., Ueda, Y.: Orbital approach to microstate free entropy. Int. J. Math. 20(2), 227–273 (2009)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Hiai, F., Ueda, Y.: Notes on microstate free entropy of projections. Publ. Res. Inst. Math. Sci. 44(1), 49–89 (2008)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Hiai, F., Ueda, Y.: A log-Sobolev type inequality for free entropy of two projections. Ann. Inst. Henri Poincaré Probab. Stat. 45(1), 239–249 (2009)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Izumi, M., Ueda, Y.: Remarks on free mutual information and orbital free entropy. Nagoya Math. J. 220, 45–66 (2015)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Johnstone, I.M.: Multivariate analysis and Jacobi ensembles: largest eigenvalue, Tracy–Widom limits and rates of convergence. Ann. Stat. 36(6), 2638–2716 (2008)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Kemp, T.: Heat kernel empirical Laws on \({\mathbb{U}}_N\) and \({\mathbb{G}}{\mathbb{L}}_N\). J. Theor. Probab. (2015, to appear)Google Scholar
  31. 31.
    Kemp, T.: The large-\(N\) limits of Brownian motions on \({\mathbb{G}L}_N\). Int. Math. Res. Not. 2016(13), 4012–4057 (2016)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Kemp, T., Nourdin, I., Peccati, G., Speicher, R.: Wigner chaos and the fourth moment. Ann. Probab. 40(4), 1577–1635 (2012)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Lévy, T.: Schur–Weyl duality and the heat kernel measure on the unitary group. Adv. Math. 218(2), 537–575 (2008)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Lévy, T., Maïda, M.: Central limit theorem for the heat kernel measure on the unitary group. J. Funct. Anal. 259(12), 3163–3204 (2010)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Male, C.: The norm of polynomials in large random and deterministic matrices. Probab. Theory Relat. Fields 154(3–4), 477–532 (2012). With an appendix by Dimitri ShlyakhtenkoGoogle Scholar
  36. 36.
    Marchenko, V.A., Pastur, L.A.: Distribution of eigenvalues in certain sets of random matrices. Mat. Sb. (N.S.) 72(114), 507–536 (1967)Google Scholar
  37. 37.
    Mehta, M.L.: Random Matrices, vol. 142, 3rd edn. Pure and Applied Mathematics (Amsterdam). Elsevier/Academic Press, Amsterdam (2004)Google Scholar
  38. 38.
    Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability, vol. 335. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2006)Google Scholar
  39. 39.
    Pólya, G., Szegő, G.: Problems and Theorems in Analysis. I. Classics in Mathematics. Springer, Berlin (1998). Series, integral calculus, theory of functions, Translated from the German by Dorothee Aeppli, Reprint of the 1978 English translationGoogle Scholar
  40. 40.
    Rains, E.M.: Combinatorial properties of Brownian motion on the compact classical groups. J. Theor. Probab. 10(3), 659–679 (1997)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Robinson, D.W.: Elliptic Operators and Lie Groups. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press/Oxford Science Publications, New York (1991)Google Scholar
  42. 42.
    Saloff-Coste, L.: Aspects of Sobolev-Type Inequalities. London Mathematical Society Lecture Note Series, vol. 289. Cambridge University Press, Cambridge (2002)Google Scholar
  43. 43.
    Sengupta, A.: Traces in Two-Dimensional QCD: The Large \(N\)-Limit. Aspects of Mathematics, E38. Friedrich Vieweg & Sohn, Wiesbaden (2008)Google Scholar
  44. 44.
    Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159(1), 151–174 (1994)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Varopoulos, N.T., Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups, vol. 100. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1992)Google Scholar
  46. 46.
    Voiculescu, D.: Limit laws for random matrices and free products. Invent. Math. 104(1), 201–220 (1991)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Voiculescu, D.: A strengthened asymptotic freeness result for random matrices with applications to free entropy. Int. Math. Res. Not. 1, 41–63 (1998)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Voiculescu, D.: The analogues of entropy and of Fisher’s information measure in free probability theory. VI. Liberation and mutual free information. Adv. Math. 146(2), 101–166 (1999)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Voiculescu, D.V., Dykema, K.J., Nica, A.: Free Random Variables. CRM Monograph Series, vol. 1. American Mathematical Society, Providence (1992). A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groupsGoogle Scholar
  50. 50.
    Wigner, E.P.: On the distribution of the roots of certain symmetric matrices. Ann. Math. 2(67), 325–327 (1958)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Benoît Collins
    • 1
    • 2
  • Antoine Dahlqvist
    • 3
  • Todd Kemp
    • 4
  1. 1.Department of MathematicsKyoto UniversityKyotoJapan
  2. 2.CNRSLyonFrance
  3. 3.Statistical LaboratoryCentre for Mathematical SciencesCambridgeUK
  4. 4.Department of MathematicsUniversity of California, San DiegoLa JollaUSA

Personalised recommendations