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Forum Mathematicum

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Volume 28, Issue 5


Singular values and evenness symmetry in random matrix theory

Folkmar Bornemann / Peter J. Forrester
  • Corresponding author
  • Department of Mathematics and Statistics, University of Melbourne, Swanston St, Parkville Victoria 3010, Australia; and ARC Centre of Excellence for Mathematical & Statistical Frontiers, University of Melbourne, Parkville VIC 3010, Australia
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Published Online: 2015-10-14 | DOI: https://doi.org/10.1515/forum-2015-0055


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.

Keywords: Random matrices; evenness symmetry; singular values; gap probabilities

MSC 2010: 15B52; 60K35


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About the article

Received: 2015-03-25

Published Online: 2015-10-14

Published in Print: 2016-09-01

Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: TRR 109

Funding Source: Australian Research Council

Award identifier / Grant number: DP140102613

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.

Citation Information: Forum Mathematicum, Volume 28, Issue 5, Pages 873–891, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2015-0055.

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