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From random matrices to random analytic functions
"... Singular points of random matrixvalued analytic functions are a common generalization of eigenvalues of random matrices and zeros of random polynomials. The setting is that we have an analytic function of z taking values in the space of n × n matrices. Singular points are those (random) z where the ..."
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Singular points of random matrixvalued analytic functions are a common generalization of eigenvalues of random matrices and zeros of random polynomials. The setting is that we have an analytic function of z taking values in the space of n × n matrices. Singular points are those (random) z where the matrix becomes singular, that is, the zeros of the determinant. This notion was introduced in the Ph.D thesis [10] of the author, where some basic facts were found. Of course, singular points are just the zeros of the (random analytic function) determinant, so in what sense is this concept novel? In case of random matrices as well as random analytic functions, the following features may be observed. 1. For very special models, usually with independent Gaussian coefficients or entries, one may solve exactly for the distribution of zeros or eigenvalues. 2. For more general models with independent coefficients or entries, under rather weak assumptions on moments, one can usually analyze the empirical measure of eigenvalues or zeros as the size of the matrix increases or the
Derivation of an eigenvalue probability density function relating to the Poincaré disk
, 906
"... A result of Zyczkowski and Sommers [J. Phys. A 33, 2045–2057 (2000)] gives the eigenvalue probability density function for the top N × N subblock 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 subblocks, in ..."
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A result of Zyczkowski and Sommers [J. Phys. A 33, 2045–2057 (2000)] gives the eigenvalue probability density function for the top N × N subblock 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 subblocks, 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 subblocks to a many body quantum state, and to the onecomponent plasma, on the pseudosphere. 1