The Annals of Probability

Local single ring theorem

Florent Benaych-Georges

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The single ring theorem, by Guionnet, Krishnapur and Zeitouni in Ann. of Math. (2) 174 (2011) 1189–1217, describes the empirical eigenvalue distribution of a large generic matrix with prescribed singular values, that is, an $N\times N$ matrix of the form $A=UTV$, with $U,V$ some independent Haar-distributed unitary matrices and $T$ a deterministic matrix whose singular values are the ones prescribed. In this text, we give a local version of this result, proving that it remains true at the microscopic scale $(\log N)^{-1/4}$. On our way to prove it, we prove a matrix subordination result for singular values of sums of non-Hermitian matrices, as Kargin did in Ann. Probab. 43 (2015) 2119–2150 for Hermitian matrices. This allows to prove a local law for the singular values of the sum of two non-Hermitian matrices and a delocalization result for singular vectors.

Article information

Source
Ann. Probab., Volume 45, Number 6A (2017), 3850-3885.

Dates
Received: February 2015
Revised: April 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1511773666

Digital Object Identifier
doi:10.1214/16-AOP1151

Mathematical Reviews number (MathSciNet)
MR3729617

Zentralblatt MATH identifier
1383.15033

Subjects
Primary: 15B52: Random matrices 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 46L54: Free probability and free operator algebras

Keywords
Random matrices single ring theorem local laws free convolution free probability theory haar measure

Citation

Benaych-Georges, Florent. Local single ring theorem. Ann. Probab. 45 (2017), no. 6A, 3850--3885. doi:10.1214/16-AOP1151. https://projecteuclid.org/euclid.aop/1511773666


Export citation

References

  • [1] Anderson, G. W., Guionnet, A. and Zeitouni, O. (2010). An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge Univ. Press, Cambridge.
  • [2] Bao, Z., Erdős, L. and Schnelli, K. (2016). Local stability of the free additive convolution. J. Funct. Anal. 271 672–719.
  • [3] Bao, Z., Erdős, L. and Schnelli, K. Local law of addition of random matrices on optimal scale. Preprint. Available at arXiv:1509.07080.
  • [4] Basak, A. and Dembo, A. (2013). Limiting spectral distribution of sums of unitary and orthogonal matrices. Electron. Commun. Probab. 18 no. 69, 19.
  • [5] Belinschi, S. T. (2008). The Lebesgue decomposition of the free additive convolution of two probability distributions. Probab. Theory Related Fields 142 125–150.
  • [6] Belinschi, S. T. (2014). $L^{\infty}$-boundedness of density for free additive convolutions. Rev. Roumaine Math. Pures Appl. 59 173–184.
  • [7] Belinschi, S. T., Benaych-Georges, F. and Guionnet, A. (2009). Regularization by free additive convolution, square and rectangular cases. Complex Anal. Oper. Theory 3 611–660.
  • [8] Belinschi, S. T. and Bercovici, H. (2007). A new approach to subordination results in free probability. J. Anal. Math. 101 357–365.
  • [9] Belinschi, S. T., Bercovici, H., Capitaine, M. and Février, M. Outliers in the spectrum of large deformed unitarily invariant models. Preprint. Available at arXiv:1207.5443.
  • [10] Benaych-Georges, F. (2009). Rectangular random matrices, related convolution. Probab. Theory Related Fields 144 471–515.
  • [11] Benaych-Georges, F. (2015). Exponential bounds for the support convergence in the single ring theorem. J. Funct. Anal. 268 3492–3507.
  • [12] Benaych-Georges, F. and Knowles, A. Lectures on the local semicircle law for Wigner matrices. Preprint. Available at arXiv:1601.04055.
  • [13] Bercovici, H. and Voiculescu, D. (1998). Regularity questions for free convolution. In Nonselfadjoint Operator Algebras, Operator Theory, and Related Topics. Oper. Theory Adv. Appl. 104 37–47. Birkhäuser, Basel.
  • [14] Biane, P. (1998). Processes with free increments. Math. Z. 227 143–174.
  • [15] Bordenave, C. and Chafaï, D. (2012). Around the circular law. Probab. Surv. 9 1–89.
  • [16] Bourgade, P., Yau, H.-T. and Yin, J. (2014). The local circular law for random matrices. Probab. Theory Related Fields 159 545–595.
  • [17] Bourgade, P., Yau, H.-T. and Yin, J. (2014). The local circular law II: The edge case. Probab. Theory Related Fields 159 619–660.
  • [18] Chafaï, D., Guédon, O., Lecué, G. and Pajor, A. (2012). Interactions Between Compressed Sensing Random Matrices and High Dimensional Geometry. Panoramas et Synthèses 37. Société Mathématique de France, Paris.
  • [19] Erdős, L., Schlein, B. and Yau, H.-T. (2009). Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices. Ann. Probab. 37 815–852.
  • [20] Erdős, L., Ramírez, J., Schlein, B. and Yau, H.-T. (2010). Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation. Electron. J. Probab. 15 526–603.
  • [21] Ferreira, O. P. and Svaiter, B. F. Kantorovich’s theorem on Newton’s method. Preprint. Available at arXiv:1209.5704.
  • [22] Guionnet, A., Krishnapur, M. and Zeitouni, O. (2011). The single ring theorem. Ann. of Math. (2) 174 1189–1217.
  • [23] Guionnet, A. and Zeitouni, O. (2012). Support convergence in the single ring theorem. Probab. Theory Related Fields 154 661–675.
  • [24] Haagerup, U. and Larsen, F. (2000). Brown’s spectral distribution measure for $R$-diagonal elements in finite von Neumann algebras. J. Funct. Anal. 176 331–367.
  • [25] Hardy, G. H. (1915). The mean value of the modulus of ananalytic function. Proc. Lond. Math. Soc. (3) 14 269–277.
  • [26] Kargin, V. (2012). A concentration inequality and a local law for the sum of two random matrices. Probab. Theory Related Fields 154 677–702.
  • [27] Kargin, V. (2013). An inequality for the distance between densities of free convolutions. Ann. Probab. 41 3241–3260.
  • [28] Kargin, V. (2015). Subordination of the resolvent for a sum of random matrices. Ann. Probab. 43 2119–2150.
  • [29] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Amer. Math. Soc., Providence, RI.
  • [30] Nica, A. and Speicher, R. (2006). Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series 335. Cambridge Univ. Press, Cambridge.
  • [31] Rudelson, M. and Vershynin, R. (2014). Invertibility of random matrices: Unitary and orthogonal perturbations. J. Amer. Math. Soc. 27 293–338.
  • [32] Tao, T. and Vu, V. (2010). Random matrices: Universality of ESDs and the circular law. Ann. Probab. 38 2023–2065.
  • [33] Voiculescu, D. V., Dykema, K. J. and Nica, A. (1992). Free Random Variables. CRM Monograph Series 1. Amer. Math. Soc., Providence, RI.
  • [34] Yin, I. (2014). The local circular law III: General case. Probab. Theory Related Fields 160 679–732.