Universal Level dynamics of Complex Systems

TL;DR

This paper investigates the universal behavior of eigenvalue distributions in large complex systems under random perturbations, showing that their evolution follows a universal pattern similar to classical random matrix ensembles.

Contribution

It demonstrates the universality of eigenvalue dynamics across different random perturbation ensembles, including non-Gaussian cases, in the large system limit.

Findings

Eigenvalue distribution evolution follows a Fokker-Planck equation similar to Gaussian cases.

Large N behavior is universal, independent of perturbation details.

Results connect generalized ensembles to classical random matrix theory.

Abstract

. We study the evolution of the distribution of eigenvalues of a $N\times N$ matrix subject to a random perturbation drawn from (i) a generalized Gaussian ensemble (ii) a non-Gaussian ensemble with a measure variable under the change of basis. It turns out that, in the case (i), a redefinition of the parameter governing the evolution leads to a Fokker-Planck equation similar to the one obtained when the perturbation is taken from a standard Gaussian ensemble (with invariant measure). This equivalence can therefore help us to obtain the correlations for various physically-significant cases modeled by generalized Gaussian ensembles by using the already known correlations for standard Gaussian ensembles. For large $N$-values, our results for both cases (i) and (ii) are similar to those obtained for Wigner-Dyson gas as well as for the perturbation taken from a standard Gaussian ensemble. This seems to suggest the independence of evolution, in thermodynamic limit, from the nature of perturbation involved as well as the initial conditions and therefore universality of dynamics of the eigenvalues of complex systems.