The Annals of Probability

Strong convergence of eigenangles and eigenvectors for the circular unitary ensemble

Kenneth Maples, Joseph Najnudel, and Ashkan Nikeghbali

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It is known that a unitary matrix can be decomposed into a product of complex reflections, one for each dimension, and that these reflections are independent and uniformly distributed on the space where they live if the initial matrix is Haar-distributed. If we take an infinite sequence of such reflections, and consider their successive products, then we get an infinite sequence of unitary matrices of increasing dimension, all of them following the circular unitary ensemble.

In this coupling, we show that the eigenvalues of the matrices converge almost surely to the eigenvalues of the flow, which are distributed according to a sine-kernel point process, and we get some estimates of the rate of convergence. Moreover, we also prove that the eigenvectors of the matrices converge almost surely to vectors which are distributed as Gaussian random fields on a countable set.

Article information

Ann. Probab., Volume 47, Number 4 (2019), 2417-2458.

Received: May 2017
Revised: September 2018
First available in Project Euclid: 4 July 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60F15: Strong theorems

Random matrix circular unitary ensemble convergence of eigenvalues convergence of eigenvectors virtual isometries complex reflections


Maples, Kenneth; Najnudel, Joseph; Nikeghbali, Ashkan. Strong convergence of eigenangles and eigenvectors for the circular unitary ensemble. Ann. Probab. 47 (2019), no. 4, 2417--2458. doi:10.1214/18-AOP1311.

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  • [1] Borodin, A. and Olshanski, G. (2001). Infinite random matrices and ergodic measures. Comm. Math. Phys. 223 87–123.
  • [2] Bourgade, P., Hughes, C. P., Nikeghbali, A. and Yor, M. (2008). The characteristic polynomial of a random unitary matrix: A probabilistic approach. Duke Math. J. 145 45–69.
  • [3] Bourgade, P., Najnudel, J. and Nikeghbali, A. (2013). A unitary extension of virtual permutations. Int. Math. Res. Not. IMRN 2013 4101–4134.
  • [4] Chhaibi, R., Najnudel, J. and Nikeghbali, A. (2017). The circular unitary ensemble and the Riemann zeta function: The microscopic landscape and a new approach to ratios. Invent. Math. 207 23–113.
  • [5] Jacod, J., Kowalski, E. and Nikeghbali, A. (2011). Mod-Gaussian convergence: New limit theorems in probability and number theory. Forum Math. 23 835–873.
  • [6] Katz, N. M. and Sarnak, P. (1999). Random Matrices, Frobenius Eigenvalues, and Monodromy. American Mathematical Society Colloquium Publications 45. Amer. Math. Soc., Providence, RI.
  • [7] Keating, J. P. and Snaith, N. C. (2000). Random matrix theory and $\zeta(1/2+it)$. Comm. Math. Phys. 214 57–89.
  • [8] Kerov, S., Olshanski, G. and Vershik, A. (1993). Harmonic analysis on the infinite symmetric group. A deformation of the regular representation. C. R. Acad. Sci. Paris Sér. I Math. 316 773–778.
  • [9] Kowalski, E. and Nikeghbali, A. (2010). Mod-Poisson convergence in probability and number theory. Int. Math. Res. Not. IMRN 2010 3549–3587.
  • [10] Mehta, M. L. (2004). Random Matrices, 3rd ed. Pure and Applied Mathematics (Amsterdam) 142. Elsevier/Academic Press, Amsterdam.
  • [11] Neretin, Y. A. (2002). Hua-type integrals over unitary groups and over projective limits of unitary groups. Duke Math. J. 114 239–266.