Electronic Communications in Probability

On the eigenvalues of truncations of random unitary matrices

Elizabeth Meckes and Kathryn Stewart

Full-text: Open access

Abstract

We consider the empirical eigenvalue distribution of an $m\times m$ principle submatrix of an $n\times n$ random unitary matrix distributed according to Haar measure. Earlier work of Petz and Réffy identified the limiting spectral measure if $\frac{m} {n}\to \alpha $, as $n\to \infty $; under suitable scaling, the family $\{\mu _{\alpha }\}_{\alpha \in (0,1)}$ of limiting measures interpolates between uniform measure on the unit disc (for small $\alpha $) and uniform measure on the unit circle (as $\alpha \to 1$). In this note, we prove an explicit concentration inequality which shows that for fixed $n$ and $m$, the bounded-Lipschitz distance between the empirical spectral measure and the corresponding $\mu _{\alpha }$ is typically of order $\sqrt{\frac {\log (m)}{m}} $ or smaller. The approach is via the theory of two-dimensional Coulomb gases and makes use of a new “Coulomb transport inequality” due to Chafaï, Hardy, and Maïda.

Article information

Source
Electron. Commun. Probab., Volume 24 (2019), paper no. 57, 12 pp.

Dates
Received: 6 December 2018
Accepted: 20 July 2019
First available in Project Euclid: 13 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1568361883

Digital Object Identifier
doi:10.1214/19-ECP258

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

Keywords
random matrices truncations submatrices empirical spectral measure Coulomb gas concentration inequalities Haar measure

Rights
Creative Commons Attribution 4.0 International License.

Citation

Meckes, Elizabeth; Stewart, Kathryn. On the eigenvalues of truncations of random unitary matrices. Electron. Commun. Probab. 24 (2019), paper no. 57, 12 pp. doi:10.1214/19-ECP258. https://projecteuclid.org/euclid.ecp/1568361883


Export citation

References

  • [1] Djalil Chafaï, Adrien Hardy, and Mylène Maïda. Concentration for Coulomb gases and Coulomb transport inequalities. arXiv:1610.00980, 2016.
  • [2] Persi Diaconis and Mehrdad Shahshahani. On the eigenvalues of random matrices. J. Appl. Probab., 31A:49–62, 1994. Studies in applied probability.
  • [3] William Feller. An introduction to probability theory and its applications. Vol. I. Third edition. John Wiley & Sons, Inc., New York-London-Sydney, 1968.
  • [4] Yan V. Fyodorov and Hans-Jürgen Sommers. Statistics of resonance poles, phase shifts and time delays in quantum chaotic scattering: random matrix approach for systems with broken time-reversal invariance. J. Math. Phys., 38(4):1918–1981, 1997. Quantum problems in condensed matter physics.
  • [5] Tiefeng Jiang. Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles. Probab. Theory Related Fields, 144(1-2):221–246, 2009.
  • [6] Elizabeth S. Meckes and Mark W. Meckes. Spectral measures of powers of random matrices. Electron. Commun. Probab., 18:no. 78, 13 pp., 2013.
  • [7] Dénes Petz and Júlia Réffy. Large deviation for the empirical eigenvalue density of truncated Haar unitary matrices. Probab. Theory Related Fields, 133(2):175–189, 2005.
  • [8] Karol Życzkowski and Hans-Jürgen Sommers. Truncations of random unitary matrices. J. Phys. A, 33(10):2045–2057, 2000.