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Licensed Unlicensed Requires Authentication Published by De Gruyter October 14, 2015

Singular values and evenness symmetry in random matrix theory

Folkmar Bornemann and Peter J. Forrester
From the journal Forum Mathematicum


Complex Hermitian random matrices with a unitary symmetry can be distinguished by a weight function. When this is even, it is a known result that the distribution of the singular values can be decomposed as the superposition of two independent eigenvalue sequences distributed according to particular matrix ensembles with chiral unitary symmetry. We give decompositions of the distribution of singular values, and the decimation of the singular values – whereby only even, or odd, labels are observed – for real symmetric random matrices with an orthogonal symmetry, and even weight. This requires further specifying the functional form of the weight to one of three types – Gauss, symmetric Jacobi or Cauchy. Inter-relations between gap probabilities with orthogonal and unitary symmetry follow as a corollary. The Gauss case has appeared in a recent work of Bornemann and La Croix. The Cauchy case, when appropriately specialised and upon stereographic projection, gives decompositions for the analogue of the singular values for the circular unitary and circular orthogonal ensembles.

MSC 2010: 15B52; 60K35

Communicated by Jörg Brüdern

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: TRR 109

Funding source: Australian Research Council

Award Identifier / Grant number: DP140102613

Funding statement: The work of FB was supported by the DFG-Collaborative Research Center, TRR 109, “Discretization in Geometry and Dynamics”. The work of PJF was supported by the Australian Research Council through the grant DP140102613.


[1] Anderson G. W., A short proof of Selberg’s generalized beta formula, Forum. Math. 3 (1991), no. 4, 415–417. 10.1515/form.1991.3.415Search in Google Scholar

[2] Bornemann F., On the numerical evaluation of distributions in random matrix theory: A review, Markov Process. Related Fields 16 (2010), no. 4, 803–866. Search in Google Scholar

[3] Bornemann F. and La Croix M., The singular values of the GOE, preprint 2015, 10.1142/S2010326315500094Search in Google Scholar

[4] de Bruijn N. G., On some multiple integrals involving determinants, J. Indian Math. Soc. (N.S.) 19 (1955), 133–151. Search in Google Scholar

[5] Dixon A. L., Generalisations of Legendre’s formula KE-(K-E)K=12π, Proc. Lond. Math. Soc. 3 (1905), 206–224. 10.1112/plms/s2-3.1.206Search in Google Scholar

[6] Dyson F. J., Statistical theory of energy levels of complex systems. III, J. Math. Phys. 3 (1962), 166–175. 10.1063/1.1703775Search in Google Scholar

[7] Edelman A. and La Croix M., The singular values of the GUE (less is more), preprint 2014, 10.1142/S2010326315500215Search in Google Scholar

[8] Forrester P. J., Evenness symmetry and inter-relationships between gap probabilities in random matrix theory, Forum Math. 18 (2006), no. 5, 711–743. 10.1515/FORUM.2006.036Search in Google Scholar

[9] Forrester P. J., A random matrix decimation procedure relating β=2/(r+1) to β=2(r+1), Comm. Math. Phys. 285 (2009), no. 2, 653–672. 10.1007/s00220-008-0616-0Search in Google Scholar

[10] Forrester P. J., Log-Gases and Random Matrices, London Math. Soc. Monogr. Ser. 34, Princeton University Press, Princeton, 2010. 10.1515/9781400835416Search in Google Scholar

[11] Forrester P. J. and Lebowitz J. L., Local central limit theorem for determinantal point processes, J. Stat. Phys. 157 (2014), no. 1, 60–69. 10.1007/s10955-014-1071-2Search in Google Scholar

[12] Forrester P. J. and Rains E. M., Interrelationships between orthogonal, unitary and symplectic matrix ensembles, Random Matrix Models and Their Applications (Berkeley 1999), Math. Sci. Res. Inst. Publ. 40, Cambridge University Press, Cambridge (2001), 171–207. Search in Google Scholar

[13] Gaudin M., Sur la loi limite de l’espacement des valeurs propres d’une matrice aléatoire, Nucl. Phys. 25 (1961), 447–458. 10.1016/0029-5582(61)90176-6Search in Google Scholar

[14] Gunson J., Proof of a conjecture of Dyson in the statistical theory of energy levels, J. Math. Phys. 3 (1962), 752–753. 10.1063/1.1724277Search in Google Scholar

[15] Mehta M. L., Power series for level spacing functions of random matrix ensembles, Z. Phys. B 86 (1992), no. 2, 285–290. 10.1007/BF01313838Search in Google Scholar

[16] Mehta M. L. and Dyson F. J., Statistical theory of the energy levels of complex systems. V, J. Math. Phys. 4 (1963), 713–719. 10.1063/1.1704009Search in Google Scholar

[17] Rains E. M., Images of eigenvalue distributions under power maps, Probab. Theory Related Fields 125 (2003), no. 4, 522–538. 10.1007/s00440-002-0250-2Search in Google Scholar

Received: 2015-3-25
Published Online: 2015-10-14
Published in Print: 2016-9-1

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