The Annals of Probability

Limits of spiked random matrices II

Alex Bloemendal and Bálint Virág

Full-text: Open access


The top eigenvalues of rank $r$ spiked real Wishart matrices and additively perturbed Gaussian orthogonal ensembles are known to exhibit a phase transition in the large size limit. We show that they have limiting distributions for near-critical perturbations, fully resolving the conjecture of Baik, Ben Arous and Péché [Duke Math. J. (2006) 133 205–235]. The starting point is a new $(2r+1)$-diagonal form that is algebraically natural to the problem; for both models it converges to a certain random Schrödinger operator on the half-line with $r\times r$ matrix-valued potential. The perturbation determines the boundary condition and the low-lying eigenvalues describe the limit, jointly as the perturbation varies in a fixed subspace. We treat the real, complex and quaternion ($\beta=1,2,4$) cases simultaneously. We further characterize the limit laws in terms of a diffusion related to Dyson’s Brownian motion, or alternatively a linear parabolic PDE; here $\beta$ appears simply as a parameter. At $\beta=2$, the PDE appears to reconcile with known Painlevé formulas for these $r$-parameter deformations of the GUE Tracy–Widom law.

Article information

Ann. Probab., Volume 44, Number 4 (2016), 2726-2769.

Received: June 2012
Revised: May 2015
First available in Project Euclid: 2 August 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)

Random matrix theory finite rank perturbations spiked model Tracy–Widom distributions BBP phase transition stochastic Airy operator


Bloemendal, Alex; Virág, Bálint. Limits of spiked random matrices II. Ann. Probab. 44 (2016), no. 4, 2726--2769. doi:10.1214/15-AOP1033.

Export citation


  • Anderson, G. W., Guionnet, A. and Zeitouni, O. (2010). An Introduction to Random Matrices. Cambridge Univ. Press, Cambridge.
  • Baik, J. (2006). Painlevé formulas of the limiting distributions for nonnull complex sample covariance matrices. Duke Math. J. 133 205–235.
  • Baik, J., Ben Arous, G. and Péché, S. (2005). Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33 1643–1697.
  • Baik, J. and Rains, E. M. (2000). Limiting distributions for a polynuclear growth model with external sources. J. Stat. Phys. 100 523–541.
  • Baik, J. and Rains, E. M. (2001). The asymptotics of monotone subsequences of involutions. Duke Math. J. 109 205–281.
  • Baik, J. and Wang, D. (2013). On the largest eigenvalue of a Hermitian random matrix model with spiked external source II: Higher rank cases. Int. Math. Res. Not. IMRN 14 3304–3370.
  • Baur, G. and Kratz, W. (1989). A general oscillation theorem for selfadjoint differential systems with applications to Sturm–Liouville eigenvalue problems and quadratic functionals. Rend. Circ. Mat. Palermo (2) 38 329–370.
  • Bloemendal, A. and Baik, J. (2013). Unpublished manuscript.
  • Bloemendal, A. and Virág, B. (2013). Limits of spiked random matrices I. Probab. Theory Related Fields 156 795–825.
  • Dumitriu, I. and Edelman, A. (2002). Matrix models for beta ensembles. J. Math. Phys. 43 5830–5847.
  • Dyson, F. J. (1962). A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3 1191–1198.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • Forrester, P. J. (2013). Probability densities and distributions for spiked and general variance Wishart $\beta$-ensembles. Random Matrices Theory Appl. 2 1350011, 19.
  • Halmos, P. R. (1951). Introduction to Hilbert Space and the Theory of Spectral Multiplicity. Chelsea, New York.
  • Johnstone, I. M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 295–327.
  • Mo, M. Y. (2012). Rank 1 real Wishart spiked model. Comm. Pure Appl. Math. 65 1528–1638.
  • Morse, M. (1932). The Calculus of Variations in the Large. American Mathematical Society Colloquium Publications 18. Amer. Math. Soc., Providence, RI. (1996 reprint of the original).
  • Morse, M. (1973). Variational Analysis: Critical Extremals and Sturmian Extensions. Interscience Publishers [Wiley], New York.
  • Ramírez, J. A., Rider, B. and Virág, B. (2011). Beta ensembles, stochastic Airy spectrum, and a diffusion. J. Amer. Math. Soc. 24 919–944.
  • Reed, M. and Simon, B. (1980). Methods of Modern Mathematical Physics. I: Functional Analysis, 2nd ed. Academic Press, Inc., New York.
  • Reid, W. T. (1971). Ordinary Differential Equations. Wiley, New York.
  • Reid, W. T. (1972). Riccati Differential Equations. Academic Press, New York.
  • Trotter, H. F. (1984). Eigenvalue distributions of large Hermitian matrices; Wigner’s semicircle law and a theorem of Kac, Murdock, and Szegő. Adv. Math. 54 67–82.
  • Wang, D. (2008). Spiked Models in Wishart Ensemble, Ph.D. thesis, Brandeis Univ. Available at arXiv:0804.0889v1.
  • Weidmann, J. (1997). Strong operator convergence and spectral theory of ordinary differential operators. Univ. Iagel. Acta Math. 34 153–163.