Explicit formulas concerning eigenvectors of weakly non-unitary matrices

DOI10.1214/22-ECP507arXiv2111.12517MaRDI QIDQ6383859

Guillaume Dubach

Publication date: 24 November 2021

Abstract: We investigate eigenvector statistics of the Truncated Unitary ensemble $mathrmTUE(N,M)$ 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 $MgeqN$, suggesting possible extensions to $mathrmTUE(N,M)$ for all values of $M$.

Mathematics Subject Classification ID

Random matrices (probabilistic aspects) (60B20) Eigenvalues, singular values, and eigenvectors (15A18) Random matrices (algebraic aspects) (15B52)

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