Tridiagonal realization of the antisymmetric Gaussian β-ensemble

Dumitriu, Ioana; Forrester, Peter J.
September 2010
Journal of Mathematical Physics;Sep2010, Vol. 51 Issue 9, p093302
Academic Journal
The Householder reduction of a member of the antisymmetric Gaussian unitary ensemble gives an antisymmetric tridiagonal matrix with all independent elements. The random variables permit the introduction of a positive parameter β, and the eigenvalue probability density function of the corresponding random matrices can be computed explicitly, as can the distribution of {qi}, the first components of the eigenvectors. Three proofs are given. One involves an inductive construction based on bordering of a family of random matrices which are shown to have the same distributions as the antisymmetric tridiagonal matrices. This proof uses the Dixon-Anderson integral from Selberg integral theory. A second proof involves the explicit computation of the Jacobian for the change of variables between real antisymmetric tridiagonal matrices, its eigenvalues, and {qi}. The third proof maps matrices from the antisymmetric Gaussian β-ensemble to those realizing particular examples of the Laguerre β-ensemble. In addition to these proofs, we note some simple properties of the shooting eigenvector and associated Prüfer phases of the random matrices.


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