Non-Universality in Random Matrix Ensembles with Soft Level Confinement

TL;DR

This paper investigates non-Gaussian random matrix ensembles with soft confinement potentials, revealing deviations from classical spectral statistics and a transition from Wigner-Dyson to Poisson behavior in certain regimes.

Contribution

It introduces new non-Gaussian RME models with long-range confining potentials and analyzes their spectral statistics through Monte Carlo simulations.

Findings

Deviations from classical SDF near the origin for power-law potentials.

Cross-over from Wigner-Dyson to Poisson in the bulk for double-logarithmic potential.

Spectral statistics depend on the form of the confining potential.

Abstract

Two families of strongly non-Gaussian random matrix ensembles (RME) are considered. They are statistically equivalent to a one-dimensional plasma of particles interacting logarithmically and confined by the potential that has the long-range behavior $V(\epsilon)\sim |\epsilon|^{\alpha}$ ($0<\alpha<1$), or $V(\epsilon)\sim \ln^{2}|\epsilon|$. The direct Monte Carlo simulations on the effective plasma model shows that the spacing distribution function (SDF) in such RME can deviate from that of the classical Gaussian ensembles. For power-law potentials, this deviation is seen only near the origin $\epsilon\sim 0$, while for the double-logarithmic potential the SDF shows the cross-over from the Wigner-Dyson to Poisson behavior in the bulk of the spectrum.