Extremal statistics of quadratic forms of GOE/GUE eigenvectors

TL;DR

This paper studies the extreme values of quadratic forms evaluated at eigenvectors of GOE/GUE matrices, showing they behave like independent Gaussians under certain conditions, with applications to invariant ensembles.

Contribution

It proves that the extremal distributions of quadratic forms of eigenvectors match Gaussian predictions when the deterministic matrix has small rank, simplifying analysis of these extremal statistics.

Findings

Extremal distributions follow Gumbel or Weibull laws depending on matrix signature.

Results hold for eigenvectors of invariant ensembles.

Asymptotic behavior matches that of independent Gaussian vectors.

Abstract

We consider quadratic forms of deterministic matrices $A$ evaluated at the random eigenvectors of a large $N \times N$ GOE or GUE matrix, or equivalently evaluated at the columns of a Haar-orthogonal or Haar-unitary random matrix. We prove that, as long as the deterministic matrix has rank much smaller than $\sqrt{N}$, the distributions of the extrema of these quadratic forms are asymptotically the same as if the eigenvectors were independent Gaussians. This reduces the problem to Gaussian computations, which we carry out in several cases to illustrate our result, finding Gumbel or Weibull limiting distributions depending on the signature of $A$. Our result also naturally applies to the eigenvectors of any invariant ensemble.