Tridiagonal realization of the anti-symmetric Gaussian $β$-ensemble

Details Tridiagonal realization of the anti-symmetric Gaussian $β$-ensemble Tridiagonal realization of the anti-symmetric Gaussian $β$-ensemble

2009-04-14

arxiv.org/abs/0904.2216v2

Physics

Mathematical Physics

22 pages; replaced with a new version containing orthogonal transformation proof for both cases (Method III)

Scientific paper

The Householder reduction of a member of the anti-symmetric Gaussian unitary ensemble gives an anti-symmetric tridiagonal matrix with all independent elements. The random variables permit the introduction of a positive parameter $\beta$, and the eigenvalue probability density function of the corresponding random matrices can be computed explicitly, as can the distribution of $\{q_i\}$, 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 anti-symmetric 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 anti-symmetric tridiagonal matrices, its eigenvalues and $\{q_i\}$. The third proof maps matrices from the anti-symmetric Gaussian $\beta$-ensemble to those realizing particular examples of the Laguerre $\beta$-ensemble. In addition to these proofs, we note some simple properties of the shooting eigenvector and associated Pr\"ufer phases of the random matrices.

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