Size Effect of Diagonal Random Matrices

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.