Truncations of Random Orthogonal Matrices

TL;DR

This paper investigates the spectral properties of truncated random orthogonal matrices, deriving exact eigenvalue densities and revealing universal behaviors in different truncation regimes.

Contribution

It provides an exact formula for eigenvalue density of truncated orthogonal matrices and characterizes their spectral behavior in various regimes.

Findings

A formula for the eigenvalue density with real eigenvalues and symmetric complex spectrum.

In the strong non-orthogonality regime, the spectrum resembles the real Ginibre ensemble.

For fixed truncation, a universal distribution of resonance widths is identified.

Abstract

Statistical properties of non--symmetric real random matrices of size $M$, obtained as truncations of random orthogonal $N\times N$ matrices are investigated. We derive an exact formula for the density of eigenvalues which consists of two components: finite fraction of eigenvalues are real, while the remaining part of the spectrum is located inside the unit disk symmetrically with respect to the real axis. In the case of strong non--orthogonality, $M/N=$const, the behavior typical to real Ginibre ensemble is found. In the case $M=N-L$ with fixed $L$, a universal distribution of resonance widths is recovered.