Extreme Eigenvalues and the Emerging Outlier in Rank-One Non-Hermitian Deformations of the Gaussian Unitary Ensemble

TY - JOUR

T1 - Extreme Eigenvalues and the Emerging Outlier in Rank-One Non-Hermitian Deformations of the Gaussian Unitary Ensemble

AU - Fyodorov, Yan V.

AU - Khoruzhenko, Boris A.

AU - Poplavskyi, Mihail

N1 - Funding Information: Y.V.F. acknowledges financial support from EPSRC Grant EP/V002473/1 “Random Hessians and Jacobians: theory and applications”. Publisher Copyright: © 2022 by the authors.

PY - 2023/1

Y1 - 2023/1

N2 - Complex eigenvalues of random matrices (Formula presented.) provide the simplest model for studying resonances in wave scattering from a quantum chaotic system via a single open channel. It is known that in the limit of large matrix dimensions (Formula presented.) the eigenvalue density of J undergoes an abrupt restructuring at (Formula presented.), the critical threshold beyond which a single eigenvalue outlier (“broad resonance”) appears. We provide a detailed description of this restructuring transition, including the scaling with N of the width of the critical region about the outlier threshold (Formula presented.) and the associated scaling for the real parts (“resonance positions”) and imaginary parts (“resonance widths”) of the eigenvalues which are farthest away from the real axis. In the critical regime we determine the density of such extreme eigenvalues, and show how the outlier gradually separates itself from the rest of the extreme eigenvalues. Finally, we describe the fluctuations in the height of the eigenvalue outlier for large but finite N in terms of the associated large deviation function.

AB - Complex eigenvalues of random matrices (Formula presented.) provide the simplest model for studying resonances in wave scattering from a quantum chaotic system via a single open channel. It is known that in the limit of large matrix dimensions (Formula presented.) the eigenvalue density of J undergoes an abrupt restructuring at (Formula presented.), the critical threshold beyond which a single eigenvalue outlier (“broad resonance”) appears. We provide a detailed description of this restructuring transition, including the scaling with N of the width of the critical region about the outlier threshold (Formula presented.) and the associated scaling for the real parts (“resonance positions”) and imaginary parts (“resonance widths”) of the eigenvalues which are farthest away from the real axis. In the critical regime we determine the density of such extreme eigenvalues, and show how the outlier gradually separates itself from the rest of the extreme eigenvalues. Finally, we describe the fluctuations in the height of the eigenvalue outlier for large but finite N in terms of the associated large deviation function.

KW - complex eigenvalues

KW - eigenvalue outlier

KW - extreme eigenvalues

KW - non-Hermitian random matrices

KW - resonance trapping

KW - resonances

UR - http://www.scopus.com/inward/record.url?scp=85146761652&partnerID=8YFLogxK

U2 - 10.3390/e25010074

DO - 10.3390/e25010074

M3 - Article

AN - SCOPUS:85146761652

SN - 1099-4300

VL - 25

JO - Entropy

JF - Entropy

IS - 1

M1 - 74

ER -