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

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.

11 pages; To appear in J.Phys A

References in corpus (6)

• Quantum conductance problems and the Jacobi ensemble
• Product of random projections, Jacobi ensembles and universality problems arising from free probability
• Zeros of Random Analytic Functions
• On absolute moments of characteristic polynomials of a certain class of complex random matrices
• A few remarks on Colour-Flavour Transformations,truncations of random unitary matrices, Berezin reproducing kernels and Selberg type integrals
• On the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values

Cited by in corpus (20)

• Around the circular law
• Universal microscopic correlation functions for products of truncated unitary matrices
• Truncations of Random Orthogonal Matrices
• Analogies between random matrix ensembles and the one-component plasma in two-dimensions
• New spectral relations between products and powers of isotropic random matrices
• One-component plasma on a spherical annulus and a random matrix ensemble
• Matrix models and eigenvalue statistics for truncations of classical ensembles of random unitary matrices
• A geometrical triumvirate of real random matrices
• Macroscopic and edge behavior of a planar jellium
• Explicit formulas concerning eigenvectors of weakly non-unitary matrices
• Winding Number Statistics of a Parametric Chiral Unitary Random Matrix Ensemble
• Winding Number Statistics for Chiral Random Matrices: Averaging Ratios of Determinants with Parametric Dependence
• Multi-Manifold Learning for Large-scale Targeted Advertising System
• Pfaffian point process for the Gaussian real generalised eigenvalue problem
• Winding Number Statistics for Chiral Random Matrices: Averaging Ratios of Parametric Determinants in the Orthogonal Case
• Exact Multivariate Amplitude Distributions for Non-Stationary Gaussian or Algebraic Fluctuations of Covariances or Correlations
• Matrix Moments in a Real, Doubly Correlated Algebraic Generalization of the Wishart Model
• Large Deviations for Random Matricial Moment Problems
• Statistical Topology -- Distribution and Density Correlations of Winding Numbers in Chiral Systems
• Winding Number Statistics for Chiral Random Matrices: Universal Correlations and Statistical Moments in the Unitary Case