Random-matrix theory

Eigenvector statistics in non-Hermitian random-matrix ensembles. The spectra of non-Hermitian random matrices have been studied in great detail.

Since the dynamics of non-Hermitian disordered systems is determined not only by the eigenvalues but also by the eigenvectors of the corresponding linear operator, we calculated eigenvector fluctuations for certain ensembles of non-Hermitian random matrices [3,4].

Our results are significant for a range of physical problems, such as random lasing media, and the decay of excited quantum systems [1,2].

Recently mathematicians became interested in our old results and extended them significantly. See for example [Fyodorov, Comm. Math. Phys. (2018)] and [Crawford & Rosenthal, arxiv:1805.08993].

[1] Statistics of resonances and nonorthogonal eigenfunctions in a model for single-channel chaotic scattering
YV Fyodorov & B Mehlig, Physical Review E 66 (2002) 045202

[2] Fano interference and cross-section fluctuations in molecular photodissociation
Y. Alhassid, Y. V. Fyodorov, T. Gorin, W. Ihra & B. Mehlig, Phys. Rev. A 73 (2006)  042711

[3] Statistical properties of eigenvectors in non-Hermitian Gaussian random matrix problems
B. Mehlig &  J. T. Chalker, J. Math. Phys. 41 (2000) 3233

[4] Eigenvector statistics in non-Hermitian random matrix ensembles
J. T. Chalker & B. Mehlig, Phys. Rev. Lett. 81 (1998) 3367