Skip to main content

On the operator norm of non-commutative polynomials in deterministic matrices and iid Haar unitary matrices

Abstract

Let \(U^N = (U_1^N,\dots ,U^N_p)\) be a p-tuple of \(N\times N\) independent Haar unitary matrices and \(Z^{NM}\) be any family of deterministic matrices in \({\mathbb {M}}_N({\mathbb {C}})\otimes {\mathbb {M}}_M({\mathbb {C}})\). Let P be a self-adjoint non-commutative polynomial. In Voiculescu (Int Math Res Notices 1:41–63, 1998), Voiculescu showed that the empirical measure of the eigenvalues of this polynomial evaluated in Haar unitary matrices and deterministic matrices converges towards a deterministic measure defined thanks to free probability theory. Now, let f be a smooth function. The main technical result of this paper is a precise bound of the difference between the expectation of

$$\begin{aligned} \frac{1}{MN} {{\,\mathrm{Tr}\,}}_{{\mathbb {M}}_N({\mathbb {C}})}\otimes {{\,\mathrm{Tr}\,}}_{{\mathbb {M}}_M({\mathbb {C}})}\left( f(P(U^N\otimes I_M,Z^{NM})) \right) , \end{aligned}$$

and its limit when N goes to infinity. If f is seven times differentiable, we show that it is bounded by \(M^2 \left\| f\right\| _{{\mathcal {C}}^6} \ln ^2(N)\times N^{-2}\). As a corollary we obtain a new proof with quantitative bounds of a result of Collins and Male which gives sufficient conditions for the operator norm of a polynomial evaluated in Haar unitary matrices and deterministic matrices to converge almost surely towards its free limit. Our result also holds in much greater generality. For instance, it allows to prove that if \(U^N\) and \(Y^{M_N}\) are independent and \(M_N=o(N^{1/3}\ln ^{-2/3}(N))\), then the norm of any polynomial in \((U^N\otimes I_{M_N}, I_N\otimes Y^{M_N})\) converges almost surely towards its free limit. Previous results required that \(M=M_N\) is constant.

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

References

  1. Anderson, G.W.: Convergence of the largest singular value of a polynomial in independent Wigner matrices. Ann. Probab. 41(3B), 2103–2181 (2013)

    Article  MathSciNet  Google Scholar 

  2. Anderson, G.W., Guionnet, A., Zeitouni, O.: An introduction to random matrices. Cambridge Studies in Advanced Mathematics, vol. 118. Cambridge University Press, Cambridge (2010)

  3. Belinschi, S., Capitaine, M.: Spectral properties of polynomials in independent Wigner and deterministic matrices. J. Funct. Anal. (2016)

  4. 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)

    Article  MathSciNet  Google Scholar 

  5. Biane, P.: Free Brownian motion, free stochastic calculus and random matrices. Am. Math. Soc. (1997)

  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)

    Article  MathSciNet  Google Scholar 

  7. Brown, N.P., Ozawa, N.: \({\cal{C}}^{*}\)-algebras and finite-dimensional approximations. In Graduate studies in mathematics. Am. Math. Soc. (2008)

  8. Cabanal-Duvillard, T.: Fluctuations de la loi empirique de grandes matrices aléatoires. Ann. Inst. H. Poincaré Probab. Statist. 37(3), 373–402 (2001)

    Article  MathSciNet  Google Scholar 

  9. Cébron, G., Kemp, T.: Fluctuations of Brownian motions on \(GL_N\). arXiv:1409.5624 (2014)

  10. Capitaine, M., Donati-Martin, C.: Strong asymptotic freeness for Wigner and Wishart matrices. Indiana Univ. Math. J. 56(2), 767–803 (2007)

    Article  MathSciNet  Google Scholar 

  11. Collins, B., Dahlqvist, A., Kemp, T.: Strong convergence of unitary Brownian motion. Probab. Theory Relat. Fields 170(1–2), 49–93 (2018)

    Article  Google Scholar 

  12. Collins, B., Guionnet, A., Parraud, F.: On the operator norm of non-commutative polynomials in deterministic matrices and iid GUE matrices (2019)

  13. Collins, B., Male, C.: The strong asymptotic freeness of Haar and deterministic matrices. Ann. Sci. Éc. Norm. Supér. 47(4), 47–163 (2014)

    MathSciNet  MATH  Google Scholar 

  14. Dabrowski, Y.: A free stochastic partial differential equation. Ann. Inst. H. Poincaré Probab. Statist. 50(4), 1404–1455 (2014)

    Article  MathSciNet  Google Scholar 

  15. Guionnet, A.: Large deviations upper bounds and central limit theorems for non-commutative functionals of Gaussian large random matrices. Ann. Inst. H. Poincaré Probab. Statist. 38(3), 341–384 (2002)

    Article  MathSciNet  Google Scholar 

  16. Guionnet, A.: Asymptotics of random matrices and related models: the uses of dyson-schwinger equations. In CBMS Regional Conference Series in Mathematics, Am. Math. Soc., (2019)

  17. Haagerup, U., Thorbjørnsen, S.: A new application of random matrices: \({\rm Ext}(C^*_{{\rm red}}({\mathbb{F}}_2))\) is not a group. Ann. Math. 162(2), 711–775 (2005)

    Article  MathSciNet  Google Scholar 

  18. Lévy, T., Maida, M.: Central limit theorem for the heat kernel measure on the unitary group. J. Funct. Anal. 259, 3163–3204 (2010)

    Article  MathSciNet  Google Scholar 

  19. Male, C.: The norm of polynomials in large random and deterministic matrices. With an appendix by Dimitri Shlyakhtenko. Probab. Theory Relat. Fields 154(3–4), 477–532 (2012)

    Article  MathSciNet  Google Scholar 

  20. Mingo, J., Śniady, P., Speicher, R.: Second order freeness and fluctuations of random matrices: II. Unit. Rand. Matrices. Adv. Math. 209(1), 212–240 (2007)

    MATH  Google Scholar 

  21. Mingo, J., Speicher, R.: Second order freeness and fluctuations of random matrices: I, Gaussian and Wishart matrices and cyclic Fock spaces. J. Funct. Anal. 235, 226–270 (2006)

    Article  MathSciNet  Google Scholar 

  22. Murphy, G.J.: C*-Algebras and operator theory. Elsevier Science, (1990)

  23. Nica, A., Speicher, R.: Lectures on the combinatorics of free probability. London Mathematical Society Lecture Note Series, vol. 335. Cambridge University Press, Cambridge (2006)

  24. Nikitopoulos, E. A.: Noncommutative \( C^k \) Functions and Fréchet Derivatives of operator functions. arXiv:2011.03126 (2020)

  25. Pisier, G.P: Random matrices and subexponential operator spaces. Israel J. Math., p. 203, (2012)

  26. Pisier, G.: Introduction to operator space theory. In London mathematical society lecture note series, Cambridge University Press (2003)

  27. Schultz, H.: Non-commutative polynomials of independent Gaussian random matrices The real and symplectic cases. Probab. Theory Related Fields 131(2), 261–309 (2005)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Voiculescu, D.: A strengthened asymptotic freeness result for random matrices with applications to free entropy. Int. Math. Res. Notices 1998(1), 41–63 (1998)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author would like to thanks his PhD supervisors Benoît Collins and Alice Guionnet for proofreading this paper and their continuous help, as well as Mikael de la Salle for its advices and the proof of Lemma 3.1. The author was partially supported by a MEXT JASSO fellowship and Labex Milyon (ANR-10-LABX-0070) of Université de Lyon.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Félix Parraud.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Parraud, F. On the operator norm of non-commutative polynomials in deterministic matrices and iid Haar unitary matrices. Probab. Theory Relat. Fields 182, 751–806 (2022). https://doi.org/10.1007/s00440-021-01101-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00440-021-01101-0

Mathematics Subject Classification

  • 60B20
  • 46L54
  • 46L09
  • 60H99