Wigner surmise for mixed symmetry classes in random matrix theory

TL;DR

This paper generalizes the Wigner surmise to mixed symmetry classes in random matrix theory, providing analytical formulas for small matrices and demonstrating their accuracy for large matrices, bridging integrable and non-integrable systems.

Contribution

It introduces a generalized Wigner surmise for mixed symmetry classes and derives analytical formulas for small matrices, applicable to large matrices, enhancing understanding of eigenvalue spacing distributions.

Findings

Analytical formulas for 2x2 and 4x4 matrices' spacing distributions.

Good approximation of large matrix distributions by small matrix formulas.

Matching coupling parameters based on local eigenvalue density.

Abstract

We consider the nearest-neighbor spacing distributions of mixed random matrix ensembles interpolating between different symmetry classes, or between integrable and non-integrable systems. We derive analytical formulas for the spacing distributions of 2x2 or 4x4 matrices and show numerically that they provide very good approximations for those of random matrices with large dimension. This generalizes the Wigner surmise, which is valid for pure ensembles that are recovered as limits of the mixed ensembles. We show how the coupling parameters of small and large matrices must be matched depending on the local eigenvalue density.