Non-Monotone Characteristic of Spectral Statistics in the Transition between Poisson and Gauss

TL;DR

This paper investigates the spectral number variances in a random matrix ensemble, revealing a non-monotone behavior due to long-range interactions and discussing implications for level repulsion and phase transitions.

Contribution

It demonstrates a non-monotone spectral variance in random matrices and links this to long-range interactions and phase transitions in level statistics.

Findings

Spectral number variances show a non-monotone increase.

The non-monotonicity arises from an overshoot in the two-level correlation.

The results suggest anti-screening of level repulsion before a phase transition.

Abstract

We have computed the spectral number variances of an extended random matrix ensemble predicted by Guhr's supersymmetry formula, showing a non-monotone increase of the curves that arises from an "overshoot" of the two-level correlation function above unity. On the basis of the most general form of $N$-level joint distribution that meets sound probabilistic conditions on matrix spaces, the above characteristic may be attributed to the {\it attractiveness} of the pair potential in long range($E >$ Thouless energy) of the underlying level gas. The approach of level dynamics indicates that the result is "anti-screening" of the level repulsion in short-range statistics of the usual random matrix prediction until the joint level distribution undergoes a phase transition (the Anderson transition).