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Local Eigenvalue Density for General MANOVA Matrices

Erdős, Lászlό and Farrell, Brendan (2013) Local Eigenvalue Density for General MANOVA Matrices. Journal of Statistical Physics, 152 (6). pp. 1003-1032. ISSN 0022-4715.

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We consider random n×n matrices of the form (XX^∗+YY^∗)^(−1/2)YY^∗(XX^∗+YY^∗)^(−1/2), where X and Y have independent entries with zero mean and variance one. These matrices are the natural generalization of the Gaussian case, which are known as MANOVA matrices and which have joint eigenvalue density given by the third classical ensemble, the Jacobi ensemble. We show that, away from the spectral edge, the eigenvalue density converges to the limiting density of the Jacobi ensemble even on the shortest possible scales of order 1/n (up to logn factors). This result is the analogue of the local Wigner semicircle law and the local Marchenko-Pastur law for general MANOVA matrices.

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Additional Information:© 2013 Springer Science+Business Media New York. Received: 14 April 2013. Accepted: 2 July 2013. Published online: 18 July 2013. L.E. was partially supported by SFB-TR12 Grant of the German Research Council. B.F. was partially supported by Joel A. Tropp under ONR awards N00014-08-1-0883 and N00014-11-1002 and a Sloan Research Fellowship.
Funding AgencyGrant Number
Deutsche Forschungsgemeinschaft (DFG)SFB-TR12
Office of Naval Research (ONR)N00014-08-1-0883
Office of Naval Research (ONR)N00014-11-1002
Alfred P. Sloan FoundationUNSPECIFIED
Subject Keywords:MANOVA random matrix; Jacobi ensemble; Local density
Issue or Number:6
Record Number:CaltechAUTHORS:20130919-143847413
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Official Citation:Erdős, L. & Farrell, B. J Stat Phys (2013) 152: 1003. doi:10.1007/s10955-013-0807-8
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:41423
Deposited By: Ruth Sustaita
Deposited On:19 Sep 2013 22:18
Last Modified:03 Oct 2019 05:48

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