Description of Quantum Systems by Random Matrix Ensembles of High Dimensions

TL;DR

This paper presents a new theorem describing the maximum of probability density functions of the second difference of energy levels in high-dimensional random matrix ensembles, highlighting phenomena like level homogenization.

Contribution

It formulates a theorem on the location of maxima of second difference distributions for various high-dimensional Gaussian ensembles and introduces concepts of level homogenization.

Findings

Maxima of second difference PDFs occur at the origin for GOE, GUE, GSE, and Poisson ensembles with N ≥ 3.

Introduces the notions of level homogenization with clustering and repulsion.

Provides a unified description of spectral fluctuations in high-dimensional quantum systems.

Abstract

The new Theorem on location of maximum of probability density functions of dimensionless second difference of the three adjacent energy levels for $N$-dimensional Gaussian orthogonal ensemble GOE($N$), $N$-dimensional Gaussian unitary ensemble GUE($N$), $N$-dimensional Gaussian symplectic ensemble GSE($N$), and Poisson ensemble PE, is formulated: {\it The probability density functions of the dimensionless second difference of the three adjacent energy levels take on maximum at the origin for the following ensembles: GOE($N$), GUE($N$), GSE($N$), and PE, where $N \geq 3$.} The notions of {\it level homogenization with level clustering} and {\it level homogenization with level repulsion} are introduced.