Explicit formulas concerning eigenvectors of weakly non-unitary matrices

TL;DR

This paper derives explicit formulas for eigenvector overlaps in the weakly non-unitary Truncated Unitary ensemble, providing insights into eigenvector statistics when only one row and column are removed.

Contribution

It offers a new explicit description of eigenvector overlaps and a method to compute their expectations conditionally on eigenvalues in the weakly non-unitary regime.

Findings

Explicit formulas for generalized overlaps as functions of eigenvalues

Method to compute expected diagonal overlaps conditionally on eigenvalues

Complementary to results in the strongly non-unitary regime

Abstract

We investigate eigenvector statistics of the Truncated Unitary ensemble $\mathrm{TUE}(N,M)$ in the weakly non-unitary case $M=1$, that is when only one row and column are removed. We provide an explicit description of generalized overlaps as deterministic functions of the eigenvalues, as well as a method to derive an exact formula for the expectation of diagonal overlaps (or squared eigenvalue condition numbers), conditionally on one eigenvalue. This complements recent results obtained in the opposite regime when $M \geq N$, suggesting possible extensions to $\mathrm{TUE}(N,M)$ for all values of $M$.