On eigenvector statistics in the spherical and truncated unitary ensembles

Abstract

We study the overlaps between right and left eigenvectors for random matrices of the spherical ensemble, as well as truncated unitary ensembles in the regime where half of the matrix at least is truncated. These two integrable models exhibit a form of duality, and the essential steps of our investigation can therefore be performed in parallel.In every case, conditionally on all eigenvalues, diagonal overlaps are shown to be distributed as a product of independent random variables with explicit distributions. 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 γ2 distribution. We also provide formulae for the conditional expectation of diagonal and off-diagonal overlaps, either with respect to one eigenvalue, or with respect to the whole spectrum. These results, analogous to what is known for the complex Ginibre ensemble, can be obtained in these cases thanks to integration techniques inspired from a previous work by Forrester & Krishnapur.

Eigenfunction non-orthogonality factors and the shape of CPA-like dips in a single-channel reflection from lossy chaotic cavities

“…First, the most prominent feature in ( 6) is the heavy-tail behaviour P (2) y (t) ∼ t −3 for t 1 rendering all moments O l nn , l 2 divergent. This tail behaviour is exactly the same as found earlier in other complex-valued non-normal random matrices [55,56,60] and seems to be the most universal feature of random diagonal overlaps. Second, the integrals P (2) y (t)t l dt can be explicitly performed for l = 0 and l = 1, reproducing for β = 2 the known mean density of the resonance widths [45] and the mean diagonal overlap [70].…”

“…Using RMT framework we derived, fully non-perturbatively, the explicit distribution of these factors for wave-chaotic scattering in systems with both broken and preserved time reversal symmetry. The results imply that O nn are heavy-tail distributed, sharing this feature with other instances of non-orthogonality factors of non-Hermitian ensembles [55,56,60,65] further supporting the claim of universality of such heavy-tail behaviour. Experimentally, statistics of O nn should be accessible within the framework of 'harmonic inversion' method [49], or via accurate study of the shape of reflection dips.…”

“…Statistics of non-orthogonal eigenvectors in non-normal random matrices is a topic of active research interest in Mathematical and Theoretical Physics starting from the seminal works [50,51]. Various aspects of non-orthogonality received a lot of attention recently [52][53][54][55][56][57][58][59][60][61][62][63][64][65]. In particular, diagonal (self )overlaps O nn are known to control sensitivity of the complex eigenvalues to perturbation of matrix entries, and κ n := √ O nn are frequently called in the mathematical literature the eigenvalue condition numbers.…”

Rank $1$ perturbations in random matrix theory -- a review of exact results

“…(2) As emphasised in [54], a noteworthy feature of (4.36) is heavy tail decay, specifically like 1/t 3 as t → ∞. This implies that all the moments O k nn diverge for k ≥ 2, as is known for the analogous quantity in the case of Ginibre type ensembles [20,51,26]. Moreover computing the moment for k = 1 reclaims (4.34).…”

Partition functions of determinantal and Pfaffian Coulomb gases with radially symmetric potentials

“…lim α→∞ Q(z) = |z| 2 /R 2 . (See [25,48] and references therein for recent works on these models.) In this case, we have (4.10)…”