Singular values and evenness symmetry in random matrix theory

Folkmar Bornemann, Peter J. Forrester

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)873-891
Number of pages19
JournalForum Mathematicum
Volume28
Issue number5
DOIs
StatePublished - 1 Sep 2016

Keywords

  • Random matrices
  • evenness symmetry
  • gap probabilities
  • singular values

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