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

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Published 2 September 2009 2009 IOP Publishing Ltd
, , Citation Peter J Forrester and Manjunath Krishnapur 2009 J. Phys. A: Math. Theor. 42 385204 DOI 10.1088/1751-8113/42/38/385204

1751-8121/42/38/385204

Abstract

A result of Zyczkowski and Sommers (2000 J. Phys. A: Math. Gen. 33 2045–57) gives the eigenvalue probability density function for the top N × N sub-block of a Haar distributed matrix from U(N + n). In the case nN, 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−1B, 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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