On invariant 2x2 \beta-ensembles of random matrices

TL;DR

This paper introduces a family of invariant 2x2 random matrix ensembles with a flexible real Dyson index, exploring their properties, connections to existing models, and potential for designing matrices with specific spectral features.

Contribution

The authors present an exactly solvable family of invariant 2x2 random matrices with a tunable real Dyson index, expanding the understanding of matrix invariance and spectral properties.

Findings

Dyson index  can be extended to any real value within an interval.

Connections established with Dumitriu-Edelman -ensembles and Poisson-Wigner crossover.

Numerical simulations confirm analytical results with high accuracy.

Abstract

We introduce and solve exactly a family of invariant 2x2 random matrices, depending on one parameter \eta, and we show that rotational invariance and real Dyson index \beta are not incompatible properties. The probability density for the entries contains a weight function and a multiple trace-trace interaction term, which corresponds to the representation of the Vandermonde-squared coupling on the basis of power sums. As a result, the effective Dyson index \beta_{eff} of the ensemble can take any real value in an interval. Two weight functions (Gaussian and non-Gaussian) are explored in detail and the connections with \beta-ensembles of Dumitriu-Edelman and the so-called Poisson-Wigner crossover for the level spacing are respectively highlighted. A curious spectral twinning between ensembles of different symmetry classes is unveiled. The proposed technical tool more generically allows for designing actual matrix models which i) are rotationally invariant; ii) have a real Dyson index \beta_{eff}; iii) have a pre-assigned confining potential or alternatively level-spacing profile. The analytical results have been checked through numerical simulations with an excellent agreement. Eventually, we discuss possible generalizations and further directions of research.