Decomposition of spectral density in individual eigenvalue contributions

Abstract

The eigenvalue densities of two random matrix ensembles, the Wigner Gaussian matrices and the Wishart covariant matrices, are decomposed in the contributions of each individual eigenvalue distribution. It is shown that the fluctuations of all eigenvalues, for medium matrix sizes, are described with a good precision by nearly normal distributions.

Spectral order statistics of Gaussian random matrices: Large deviations for trapped fermions and associated phase transitions

“…up to O(1) corrections for large N , in agreement with earlier results in both cases [17,18,29,30,37]. These estimates are respectively valid for N → ∞ such that N ρ 3 → ∞, i.e.…”

Structure of trajectories of complex-matrix eigenvalues in the Hermitian–non-Hermitian transition

“…Recently, the order statistics problem of obtaining the distribution of all eigenvalues considered an ordered sequence of random variables has been addressed [13]. Here we are interested in studying how that ordering on the real axis reflects on the eigenvalue trajectories in the complex plane as the Hermitian condition is progressively removed and also in the final eigenvalue distribution.…”

Random matrices applications to soft spectra

“…Actually, the configuration exhibited by the RMT eigenvalues as a consequence of the repulsion among them has already been considered as a crystal lattice structure. The picture is that the eigenvalues behave as a picket fence in which they vibrate around fixed points [13]. Here the analogy is extended to show that a new phase appears when spectra are subjected to an external source of randomness.…”