On Eigenvalues of the sum of two random projections

TL;DR

This paper investigates the eigenvalue distribution of the sum of two large random orthogonal projections, revealing universal local behaviors governed by classical kernels and identifying exceptional non-universal cases.

Contribution

It establishes the connection between the eigenvalues of the sum of two random projections and the Jacobi ensemble, and characterizes their universal local spectral behavior.

Findings

Eigenvalues follow sine, Bessel, or Airy kernels depending on the spectrum region.

Universal behavior is confirmed in the bulk and at the edges for large N.

An exceptional case exhibits non-universal eigenvalue behavior.

Abstract

We study the behavior of eigenvalues of matrix P_N + Q_N where P_N and Q_N are two N -by-N random orthogonal projections. We relate the joint eigenvalue distribution of this matrix to the Jacobi matrix ensemble and establish the universal behavior of eigenvalues for large N. The limiting local behavior of eigenvalues is governed by the sine kernel in the bulk and by either the Bessel or the Airy kernel at the edge depending on parameters. We also study an exceptional case when the local behavior of eigenvalues of P_N + Q_N is not universal in the usual sense.