Size Effect of Diagonal Random Matrices

TL;DR

This paper investigates how the spectral statistics of diagonal random matrices evolve with size, showing a transition from Gaussian to Poisson distribution as matrix size increases, indicating a shift from regular to integrable system behavior.

Contribution

It provides a numerical analysis of the size-dependent transition in spectral statistics of diagonal random matrices, highlighting the emergence of Poissonian behavior at large sizes.

Findings

Level spacing distribution transitions from Gaussian to Poisson as N increases.

Transition occurs around matrix size N ≈ 20.

Number variance approaches Poisson prediction with increasing N.

Abstract

The statistical distribution of levels of an integrable system is claimed to be a Poisson distribution. In this paper, we numerically generate an ensemble of N dimensional random diagonal matrices as a model for regular systems. We evaluate the corresponding nearest-neighbor spacing (NNS) distribution, which characterizes the short range correlation between levels. To characterize the long term correlations, we evaluate the level number variance. We show that, by increasing the size of matrices, the level spacing distribution evolves from the Gaussian shape that characterizes ensembles of 2\times2 matrices tending to the Poissonian as N \rightarrow \infty. The transition occurs at N \approx 20. The number variance also shows a gradual transition towards the straight line behavior predicted by the Poisson statistics.