Electronic Journal of Probability

Large complex correlated Wishart matrices: the Pearcey kernel and expansion at the hard edge

Walid Hachem, Adrien Hardy, and Jamal Najim

Full-text: Open access

Abstract

We study the eigenvalue behaviour of large complex correlated Wishart matrices near an interior point of the limiting spectrum where the density vanishes (cusp point), and refine the existing results at the hard edge as well. More precisely, under mild assumptions for the population covariance matrix, we show that the limiting density vanishes at generic cusp points like a cube root, and that the local eigenvalue behaviour is described by means of the Pearcey kernel if an extra decay assumption is satisfied. As for the hard edge, we show that the density blows up like an inverse square root at the origin. Moreover, we provide an explicit formula for the $1/N$ correction term for the fluctuation of the smallest random eigenvalue.

Article information

Source
Electron. J. Probab. Volume 21, Number (2016), paper no. 1, 36 pp.

Dates
Received: 2 January 2013
Accepted: 17 December 2014
First available in Project Euclid: 3 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1454514661

Digital Object Identifier
doi:10.1214/15-EJP4441

Subjects
Primary: 15A52
Secondary: 15A18: Eigenvalues, singular values, and eigenvectors 60F15: Strong theorems

Keywords
large random matrices Wishart matrix Pearcey kernel Bessel kernel

Rights
Creative Commons Attribution 4.0 International License.

Citation

Hachem, Walid; Hardy, Adrien; Najim, Jamal. Large complex correlated Wishart matrices: the Pearcey kernel and expansion at the hard edge. Electron. J. Probab. 21 (2016), paper no. 1, 36 pp. doi:10.1214/15-EJP4441. https://projecteuclid.org/euclid.ejp/1454514661


Export citation

References

  • [1] M. Adler, M. Cafasso, and P. van Moerbeke, Non-linear PDEs for gap probabilities in random matrices and KP theory, Phys. D 241 (2012), no. 23-24, 2265–2284.
  • [2] G. W. Anderson, A. Guionnet, and O. Zeitouni, An introduction to random matrices, Cambridge Studies in Advanced Mathematics, vol. 118, Cambridge University Press, Cambridge, 2010.
  • [3] J. Baik, G. Ben Arous, and S. Péché, Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices, Ann. Probab. 33 (2005), no. 5, 1643–1697.
  • [4] M. Bertola and M. Cafasso, The transition between the gap probabilities from the Pearcey to the airy process—a Riemann-Hilbert approach, Int. Math. Res. Not. IMRN (2012), no. 7, 1519–1568.
  • [5] P. Bleher and A. Kuijlaars, Large $n$ limit of Gaussian random matrices with external source. III. Double scaling limit, Comm. Math. Phys. 270 (2007), no. 2, 481–517.
  • [6] F. Bornemann, A note on the expansion of the smallest eigenvalue distribution of the LUE at the hard edge, arXiv:1504.00235.
  • [7] E. Brézin and S. Hikami, Level spacing of random matrices in an external source, Phys. Rev. E (3) 58 (1998), no. 6, part A, 7176–7185.
  • [8] E. Brézin and S. Hikami, Universal singularity at the closure of a gap in a random matrix theory, Phys. Rev. E (3) 57 (1998), no. 4, 4140–4149.
  • [9] M. Capitaine, S. Péché, Fluctuations at the edge of the spectrum of the full rank deformed GUE, arXiv:1402.2262.
  • [10] E. Duse, A. Metcalfe, Asymptotic Geometry of Discrete Interlaced Patterns: Part I, arXiv:1412.6653.
  • [11] E. Duse and A. Metcalfe, Universal edge fluctuations of discrete interlaced particle systems. In preparation.
  • [12] A. Edelman, A. Guionnet and S. Péché, Beyond universality in random matrix theory, arXiv:1405.7590.
  • [13] N. El Karoui, Tracy-Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices, Ann. Probab. 35 (2007), no. 2, 663–714.
  • [14] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions. Vol. II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953, Based, in part, on notes left by Harry Bateman.
  • [15] P. J. Forrester, The spectrum edge of random matrix ensembles, Nuclear Phys. B 402 (1993), no. 3, 709–728.
  • [16] I. Gohberg, S. Goldberg, and N. Krupnik, Traces and determinants of linear operators, Operator Theory: Advances and Applications, vol. 116, Birkhäuser Verlag, Basel, 2000.
  • [17] W. Hachem, A. Hardy and J. Najim, Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edges, to appear in Ann. Probab., arXiv:1409.7548.
  • [18] W. Hachem, A. Hardy and J. Najim, A survey on the eigenvalues local behaviour of large complex correlated Wishart matrices, ESAIM: Proceedings and Surveys 51 (2015), 150–174. A. Garivier et al, Editors.
  • [19] J. B. Hough, M. Krishnapur, Y. Peres, and B. Virág, Determinantal processes and independence, Probab. Surv. 3 (2006), 206–229.
  • [20] K. Johansson, Shape fluctuations and random matrices, Comm. Math. Phys. 209 (2000), no. 2, 437–476.
  • [21] K. Johansson, Random matrices and determinantal processes, Mathematical statistical physics, Elsevier B. V., Amsterdam, 2006, pp. 1–55.
  • [22] I. M. Johnstone, On the distribution of the largest eigenvalue in principal components analysis, Ann. Statist. 29 (2001), no. 2, 295–327.
  • [23] A. Knowles and J. Yin, Anisotropic local laws for random matrices, arXiv:1410.3516.
  • [24] J.O. Lee and K. Schnelli, Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population, arXiv:1409.4979.
  • [25] V. A. Marčenko and L. A. Pastur, The spectrum of random matrices, Teor. Funkciĭ Funkcional. Anal. i Priložen. Vyp. 4 (1967), 122–145.
  • [26] M. Y. Mo, Universality in complex Wishart ensembles for general covariance matrices with 2 distinct eigenvalues, J. Multivariate Anal. 101 (2010), no. 5, 1203–1225.
  • [27] A. Okounkov and N. Reshetikhin, Random skew plane partitions and the Pearcey process, Comm. Math. Phys. 269 (2007), no. 3, 571–609.
  • [28] A. Onatski, The Tracy-Widom limit for the largest eigenvalues of singular complex Wishart matrices, Ann. Appl. Probab. 18 (2008), no. 2, 470–490.
  • [29] A. Perret, G. Schehr, Finite N corrections to the limiting distribution of the smallest eigenvalue of Wishart complex matrices, arXiv:1506.02387.
  • [30] W. Rudin, Real and complex analysis, third ed., McGraw-Hill Book Co., New York, 1987.
  • [31] J. W. Silverstein and S. Choi, Analysis of the limiting spectral distribution of large-dimensional random matrices, J. Multivariate Anal. 54 (1995), no. 2, 295–309.
  • [32] B. Simon, Trace ideals and their applications, second ed., Mathematical Surveys and Monographs, vol. 120, American Mathematical Society, Providence, RI, 2005.
  • [33] C. A. Tracy and H. Widom, Level spacing distributions and the Bessel kernel, Comm. Math. Phys. 161 (1994), no. 2, 289–309.
  • [34] C. A. Tracy and H. Widom, The Pearcey process, Comm. Math. Phys. 263 (2006), no. 2, 381–400.
  • [35] J. Wishart, The generalised product moment distribution in samples from a normal multivariate population, Biometrika, 20A (1928), no. 1-2, 32–52.