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

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 x N sub-block of a Haar distributed matrix from U(N + n). In the case n ≥ 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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