World Scientific
  • Search
Skip main navigation

Cookies Notification

We use cookies on this site to enhance your user experience. By continuing to browse the site, you consent to the use of our cookies. Learn More

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

Existing users will be able to log into the site and access content. However, E-commerce and registration of new users may not be available for up to 12 hours.
For online purchase, please visit us again. Contact us at [email protected] for any enquiries.

On the analytic structure of second-order non-commutative probability spaces and functions of bounded Fréchet variation by:0 (Source: Crossref)

    In this paper, we propose a new approach to the central limit theorem (CLT) based on functions of bounded Fréchet variation for the continuously differentiable linear statistics of random matrix ensembles which relies on a weaker form of a large deviation principle for the operator norm; a Poincaré-type inequality for the linear statistics; and the existence of a second-order limit distribution. This approach frames into a single setting many known random matrix ensembles and as a consequence, classical central limit theorems for linear statistics are recovered and new ones are established, e.g. the CLT for the continuously differentiable linear statistics of block Gaussian matrices. In addition, our main results contribute to the understanding of the analytical structure of second-order non-commutative probability spaces. On the one hand, they pinpoint the source of the unbounded nature of the bilinear functional associated to these spaces; on the other hand, they lead to a general archetype for the integral representation of the second-order Cauchy transform, G2. Furthermore, we establish that the covariance of resolvents converges to this transform and that the limiting covariance of analytic linear statistics can be expressed as a contour integral in G2.

    AMSC: 60B20, 15B52, 46L54


    • 1. G. Anderson, Convergence of the largest singular value of a polynomial in independent Wigner matrices, Ann. Probab. 41 (2013) 2103–2181. CrossrefGoogle Scholar
    • 2. G. Anderson and B. Farrell, Asymptotically liberating sequences of random unitary matrices, Adv. Math. 255 (2014) 381–413. CrossrefGoogle Scholar
    • 3. G. Anderson and O. Zeitouni, A CLT for a band matrix model. Probab. Theory Related Fields 134 (2006) 283–338. CrossrefGoogle Scholar
    • 4. G. B. Arous and A. Guionnet, Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy, Probab. Theory Related Fields 108(4) (1997) 517–542. CrossrefGoogle Scholar
    • 5. Z. Bai and J. Silverstein, CLT for linear spectral statistics of large-dimensional sample covariance matrices, Ann. Probab. 32 (2004) 533–605. CrossrefGoogle Scholar
    • 6. Z. Bao, L. Erdős and K. Schnelli, Local law of addition of random matrices on optimal scale, Comm. Math. Phys. 349(3) (2017) 947–990. CrossrefGoogle Scholar
    • 7. S. Belinschi, H. Bercovici, M. Capitaine and M. Fevrier, Outliers in the spectrum of large deformed unitarily invariant models, Ann. Probab. 45(6A) (2017) 3571–3625. CrossrefGoogle Scholar
    • 8. S. Belinschi, T. Mai and R. Speicher, Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem, J. Reine Angew. Math. 732 (2017) 21–53. CrossrefGoogle Scholar
    • 9. M. Capitaine and C. Donati-Martin, Spectrum of deformed random matrices and free probability, preprint (2016), arXiv:1607.05560. Google Scholar
    • 10. B. Collins and C. Male, The strong asymptotic freeness of Haar and deterministic matrices, Ann. Sci. Éc. Norm. Supér. (4) 47(1) (2014) 147–163. CrossrefGoogle Scholar
    • 11. B. Collins, J. A. Mingo, P. Śniady and R. Speicher, Second order freeness and fluctuations of random matrices. III. Higher order freeness and free cumulants, Doc. Math. 12 (2007) 1–70. CrossrefGoogle Scholar
    • 12. P. Diaconis and M. Shahshahani, On the eigenvalues of random matrices, J. Appl. Probab. 31 (1994) 49–62. CrossrefGoogle Scholar
    • 13. M. Diaz, A. Jaramillo and J. C. Pardo, Fluctuations for matrix-valued Gaussian processes, Ann. Inst. Henri Poincaré Probab. Stat., accepted for publication. Google Scholar
    • 14. M. Diaz, J. A. Mingo and S. T. Belinschi, On the global fluctuations of block Gaussian matrices, Probab. Theory Related Fields 176(1) (2020) 599–648. CrossrefGoogle Scholar
    • 15. I. Dumitriu and A. Edelman, Global spectrum fluctuations for the β-hermite and β-Laguerre ensembles via matrix models, J. Math. Phys. 47 (2006) 063302. CrossrefGoogle Scholar
    • 16. M. Fréchet, Sur les fonctionnelles bilinéaires, Trans. Amer. Math. Soc. 16(3) (1915) 215–234. Google Scholar
    • 17. U. Haagerup and S. Thorbjornsen, A new application of random matrices: Ext(Cred(f2)) is not a group, Ann. of Math. 162 (2005) 711–775. CrossrefGoogle Scholar
    • 18. T. Hildebrandt, Introduction to the Theory of Integration, Pure and Applied Mathematics, Vol. XIII (Academic Press, New York, 1963). Google Scholar
    • 19. K. Johansson, On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J. 91(1) (1998) 151–204. CrossrefGoogle Scholar
    • 20. M. Ledoux and B. Rider, Small deviations for beta ensembles, Electron. J. Probab. 15(41) (2010) 1319–1343. Google Scholar
    • 21. M. Maïda, Large deviations for the largest eigenvalue of rank one deformations of Gaussian ensembles, Electron. J. Probab. 12 (2007) 1131–1150. CrossrefGoogle Scholar
    • 22. C. Male, The norm of polynomials in large random and deterministic matrices, Probab. Theory Related Fields 154(3–4) (2012) 477–532, With an appendix by Dimitri Shlyakhtenko. CrossrefGoogle Scholar
    • 23. V. Marchenko and L. Pastur, Distribution of eigenvalues for some sets of random matrices, Math. USSR Sb. 1(4) (1967) 457–483. CrossrefGoogle Scholar
    • 24. J. A. Mingo and A. Nica, Annular non-crossing permutations and partitions, and second-order asymptotics for random matrices, Int. Math. Res. Not. 28 (2004) 1413–1460. CrossrefGoogle Scholar
    • 25. J. A. Mingo and M. Popa, Freeness and the transposes of unitarily invariant random matrices, J. Funct. Anal. 271(4) (2016) 883–921. CrossrefGoogle Scholar
    • 26. J. A. Mingo, P. Śniady and R. Speicher, Second order freeness and fluctuations of random matrices. II. Unitary random matrices, Adv. Math. 209(1) (2007) 212–240. CrossrefGoogle Scholar
    • 27. J. A. Mingo and R. Speicher, Second order freeness and fluctuations of random matrices. I. Gaussian and Wishart matrices and cyclic Fock spaces, J. Funct. Anal. 235(1) (2006) 226–270. CrossrefGoogle Scholar
    • 28. J. A. Mingo and R. Speicher, Free Probability and Random Matrices, Fields Institute Monographs, Vol. 35 (Springer-Verlag, New York, 2017). CrossrefGoogle Scholar
    • 29. M. Morse and W. Transue, Functionals of bounded Fréchet variation, Canad. J. Math. 1 (1949) 153–165. CrossrefGoogle Scholar
    • 30. M. Morse and W. Transue, Integral representations of bilinear functionals, Proc. Natl. Acad. Sci. USA 35 (1949) 136–143. CrossrefGoogle Scholar
    • 31. L. Pastur and M. Shcherbina, Eigenvalue Distribution of Large Random Matrices, Mathematical Surveys and Monographs, Vol. 171 (American Mathematical Society, Providence, RI, 2011). CrossrefGoogle Scholar
    • 32. V. Scheidemann, Introduction to Complex Analysis in Several Variables (Birkhäuser Verlag, Basel, 2005). Google Scholar
    • 33. D. Shlyakhtenko, Free probability of type B and asymptotics of finite-rank perturbations of random matrices, Indiana Univ. Math. J. 67 (2018) 971–991. CrossrefGoogle Scholar
    • 34. D. Voiculescu, Limit laws for random matrices and free products, Invent. Math. 104(1) (1991) 201–220. CrossrefGoogle Scholar
    • 35. D. Voiculescu, A strengthened asymptotic freeness result for random matrices with applications to free entropy, Int. Math. Res. Not. 1998(1) (1998) 41–63. CrossrefGoogle Scholar
    • 36. E. Wigner, Characteristic vectors of bordered matrices with infinite dimensions, Ann. of Math. (2) 62 (1955) 548–564. CrossrefGoogle Scholar
    • 37. E. Wigner, Characteristic vectors of bordered matrices with infinite dimensions. II, Ann. of Math. (2) 65 (1957) 203–207. CrossrefGoogle Scholar
    • 38. E. Wigner, On the distribution of the roots of certain symmetric matrices, Ann. of Math. (2) 67 (1958) 325–327. CrossrefGoogle Scholar