# On eigenvector statistics in the spherical and truncated unitary ensembles

## TL;DR

This paper analyzes eigenvector overlaps in spherical and truncated unitary ensembles, revealing their distributional properties and convergence behavior, similar to the complex Ginibre ensemble, with explicit formulas for expectations.

## Contribution

It provides new results on the distribution and convergence of eigenvector overlaps in these ensembles, extending known results from the Ginibre ensemble.

## Key findings

- Diagonal overlaps are distributed as products of independent variables.
- Scaled diagonal overlaps converge to an inverse gamma distribution.
- Explicit formulas for conditional expectations of overlaps.

## Abstract

We study the overlaps between right and left eigenvectors for random matrices of the spherical and truncated unitary ensembles. Conditionally on all eigenvalues, diagonal overlaps are shown to be distributed as a product of independent random variables. This enables us to prove that the scaled diagonal overlaps, conditionally on one eigenvalue, converge in distribution to a heavy-tail limit, namely, the inverse of a $\gamma_2$ distribution. These results are analogous to what is known for the complex Ginibre ensemble. We also provide formulae for the conditional expectation of diagonal and off-diagonal overlaps, with respect to all eigenvalues.