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arXiv · 0906.5223

Derivation of an eigenvalue probability density function relating to the Poincare disk

Peter J. Forrester·Manjunath Krishnapur
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Abstract

A result of Zyczkowski and Sommers [J.Phys.A, 33, 2045--2057 (2000)] gives the eigenvalue probability density function for the top N x N sub-block of a Haar distributed matrix from U(N+n). In the case n \ge N, we rederive this result, starting from knowledge of the distribution of the sub-blocks, introducing the Schur decomposition, and integrating over all variables except the eigenvalues. The integration is done by identifying a recursive structure which reduces the dimension. This approach is inspired by an analogous approach which has been recently applied to determine the eigenvalue probability density function for random matrices A^{-1} B, where A and B are random matrices with entries standard complex normals. We relate the eigenvalue distribution of the sub-blocks to a many body quantum state, and to the one-component plasma, on the pseudosphere.

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Peter J. Forrester, Manjunath Krishnapur. 2009-09-10. Derivation of an eigenvalue probability density function relating to the Poincare disk. https://doi.org/10.1088/1751-8113%2F42%2F38%2F385204

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