Skip to main content

The spectral edge of unitary Brownian motion

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2

References

  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)

  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 author

  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)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  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)

  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)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  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)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  7. Biane, P., Speicher, R.: Free diffusions, free entropy and free Fisher information. Ann. Inst. H. PoincarĆ© Probab. Stat. 37(5), 581–606 (2001)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  8. CĆ©bron, G.: Free convolution operators and free Hall transform. J. Funct. Anal. 265(11), 2645–2708 (2013)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  9. CƩbron, G., Kemp, T.: Fluctuations of Brownian motions on \({\mathbb{G}L}_N\) (2014). arXiv:1409.5624

  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)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  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)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  12. Collins, B., Kemp, T.: Liberation of projections. J. Funct. Anal. 266(4), 1988–2052 (2014)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  13. Collins, B., Male, C.: The strong asymptotic freeness of Haar and deterministic matrices. Ann. Sci. Ɖc. Norm. SupĆ©r. (4) 47 1, 147–163 (2014)

  14. Dahlqvist, A.: Integration formula for Brownian motion on classical compact Lie groups. Ann. Inst. H. PoincarƩ Probab. Stat. (to appear)

  15. Demni, N.: Free Jacobi process. J. Theor. Probab. 21(1), 118–143 (2008)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  16. Demni, N.: Free Jacobi process associated with one projection: local inverse of the flow (2015). arXiv:1502.00013

  17. Demni, N., Hamdi, T., Hmidi, T.: Spectral distribution of the free Jacobi process. Indiana Univ. Math. J. 61(3), 1351–1368 (2012)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  18. Demni, N., Hmidi, T.: Spectral distribution of the free Jacobi process associated with one projection. Colloq. Math. 137(2), 271–296 (2014)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  19. Demni, N., Zani, M.: Large deviations for statistics of the Jacobi process. Stoch. Process. Appl. 119(2), 518–533 (2009)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  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)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  21. Erdős, L., Farrell, B.: Local eigenvalue density for general MANOVA matrices. J. Stat. Phys. 152(6), 1003–1032 (2013)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  22. Evans, L.C.: Partial Differential Equations, vol. 19, 2nd edn. Graduate Studies in Mathematics. American Mathematical Society, Providence (2010)

  23. Gustavsson, J.: Gaussian fluctuations of eigenvalues in the GUE. Ann. Inst. H. PoincarĆ© Probab. Stat. 41(2), 151–178 (2005)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  24. Haagerup, U., ThorbjĆørnsen, 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)

  25. Hiai, F., Miyamoto, T., Ueda, Y.: Orbital approach to microstate free entropy. Int. J. Math. 20(2), 227–273 (2009)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  26. Hiai, F., Ueda, Y.: Notes on microstate free entropy of projections. Publ. Res. Inst. Math. Sci. 44(1), 49–89 (2008)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  27. Hiai, F., Ueda, Y.: A log-Sobolev type inequality for free entropy of two projections. Ann. Inst. Henri PoincarĆ© Probab. Stat. 45(1), 239–249 (2009)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  28. Izumi, M., Ueda, Y.: Remarks on free mutual information and orbital free entropy. Nagoya Math. J. 220, 45–66 (2015)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  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)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  30. Kemp, T.: Heat kernel empirical Laws on \({\mathbb{U}}_N\) and \({\mathbb{G}}{\mathbb{L}}_N\). J. Theor. Probab. (2015, to appear)

  31. Kemp, T.: The large-\(N\) limits of Brownian motions on \({\mathbb{G}L}_N\). Int. Math. Res. Not. 2016(13), 4012–4057 (2016)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  32. Kemp, T., Nourdin, I., Peccati, G., Speicher, R.: Wigner chaos and the fourth moment. Ann. Probab. 40(4), 1577–1635 (2012)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  33. LĆ©vy, T.: Schur–Weyl duality and the heat kernel measure on the unitary group. Adv. Math. 218(2), 537–575 (2008)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  34. LĆ©vy, T., MaĆÆda, M.: Central limit theorem for the heat kernel measure on the unitary group. J. Funct. Anal. 259(12), 3163–3204 (2010)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  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 Shlyakhtenko

  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)

  37. Mehta, M.L.: Random Matrices, vol. 142, 3rd edn. Pure and Applied Mathematics (Amsterdam). Elsevier/Academic Press, Amsterdam (2004)

  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)

  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 translation

  40. Rains, E.M.: Combinatorial properties of Brownian motion on the compact classical groups. J. Theor. Probab. 10(3), 659–679 (1997)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  41. Robinson, D.W.: Elliptic Operators and Lie Groups. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press/Oxford Science Publications, New York (1991)

  42. Saloff-Coste, L.: Aspects of Sobolev-Type Inequalities. London Mathematical Society Lecture Note Series, vol. 289. Cambridge University Press, Cambridge (2002)

  43. Sengupta, A.: Traces in Two-Dimensional QCD: The Large \(N\)-Limit. Aspects of Mathematics, E38. Friedrich Vieweg & Sohn, Wiesbaden (2008)

  44. Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159(1), 151–174 (1994)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  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)

  46. Voiculescu, D.: Limit laws for random matrices and free products. Invent. Math. 104(1), 201–220 (1991)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  47. Voiculescu, D.: A strengthened asymptotic freeness result for random matrices with applications to free entropy. Int. Math. Res. Not. 1, 41–63 (1998)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  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)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  49. Voiculescu, D.V., Dykema, K.J., Nica, A.: Free Random Variables. CRM Monograph Series, vol.Ā 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 groups

  50. Wigner, E.P.: On the distribution of the roots of certain symmetric matrices. Ann. Math. 2(67), 325–327 (1958)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

Download references

Acknowledgements

This paper was initiated during a research visit of BC to TK in late 2012; substantial subsequent progress was made during a one-month visit of AD to BC at AIMR, Sendai during the fall of 2013. We would like to express our great thanks to UC San Diego and AIMR for their warm hospitality, fruitful work environments, and financial support that aided in the completion of this work. We would like to thank Ioana Dumitriu for helping to design the simulations that produced the figures. Finally, we express our gratitude to the anonymous referees whose careful reading and detailed comments significantly improved the presentation of this manuscript (particularly in Sect.Ā 3.2).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Todd Kemp.

Additional information

B. Collins was supported in part by Kakenhi, an NSERC discovery grant, Ontario’s ERA and ANR-14-CE25-0003.

A. Dahlqvist was supported in part by RTG 1845 and EPSRC Grant EP/I03372X/1.

T. Kemp was supported in part by NSF CAREER Award DMS-1254807.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Collins, B., Dahlqvist, A. & Kemp, T. The spectral edge of unitary Brownian motion. Probab. Theory Relat. Fields 170, 49–93 (2018). https://doi.org/10.1007/s00440-016-0753-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00440-016-0753-x

Mathematics Subject Classification

  • 15B52
  • 46L54
  • 60F15
  • 60G55
  • 60J65