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

TL;DR

This paper rederives the eigenvalue probability density function for sub-blocks of Haar-distributed matrices, connecting it to quantum states and plasma models on the Poincare disk, using a recursive integration approach.

Contribution

It introduces a new recursive method to derive eigenvalue distributions of sub-blocks of Haar matrices, linking random matrix theory to quantum and plasma models.

Findings

Derived eigenvalue density for sub-blocks with n ≥ N

Connected eigenvalue distributions to quantum states and plasma models

Provided a recursive integration technique for these distributions

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.