# Eigenvalue Statistics for Generalized Symmetric and Hermitian Matrices

## TL;DR

This paper analyzes the eigenvalue spacing distributions in generalized symmetric and Hermitian matrices, revealing crossover behaviors from clustering to repulsion and providing phase diagrams for large matrices.

## Contribution

It introduces a method to compute NNS distributions for generalized matrices and extends analysis to larger sizes, showing phase transitions between different eigenvalue spacing regimes.

## Key findings

- NNS distributions exhibit crossover from clustering to repulsion.
- Distribution analysis extends to 3x3 matrices and Hermitian matrices.
- Phase diagrams characterize regimes based on matrix variances.

## Abstract

The Nearest Neighbour Spacing (NNS) distribution can be computed for generalized symmetric 2x2 matrices having different variances in the diagonal and in the off-diagonal elements. Tuning the relative value of the variances we show that the distributions of the level spacings exhibit a crossover from clustering to repulsion as in GOE. The analysis is extended to 3x3 matrices where distributions of NNS as well as Ratio of Nearest Neighbour Spacing (RNNS) show similar crossovers. We show that it is possible to calculate NNS distributions for Hermitian matrices (N=2, 3) where also crossovers take place between clustering and repulsion as in GUE. For large symmetric and Hermitian matrices we use interpolation between clustered and repulsive regimes and identify phase diagrams with respect to the variances.

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Source: https://tomesphere.com/paper/1906.01023