Extreme eigenvalues and the emerging outlier in rank-one non-Hermitian deformations of the Gaussian Unitary Ensemble

Autor: Fyodorov, Yan V., Khoruzhenko, Boris A., Poplavskyi, Mihail
Rok vydání: 2022
Předmět:
Zdroj: Entropy 2023, 25(1), 74
Druh dokumentu: Working Paper
DOI: 10.3390/e25010074
Popis: Complex eigenvalues of random matrices $J=\text{GUE }+ i\gamma \diag (1, 0, \ldots, 0)$ 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 $N\gg 1$ the eigenvalue density of $J$ undergoes an abrupt restructuring at $\gamma = 1$, 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 $\gamma=1$ 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.
Comment: Revised version: Introduction is expanded to include more details and comparison of our results to the recent work of Dubach and Erdos; the discussion of universality of the exponent -1/3 in Section 2 clarifies how how this exponent is linked to the shape of the eigenvalue density of the undeformed matrix, several references and acknowledgements are added
Databáze: arXiv