On invariant 2x2 β-ensembles of random matrices

Abstract

We introduce and solve exactly a family of invariant 2x2 random matrices, depending on one parameter η, and we show that rotational invariance and real Dyson index βare 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 β_{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 β-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 β_{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.

Minor modifications. Published version

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• The probability distribution of spectral moments for the Gaussian beta-ensembles
• Interactions Between Different Birds of Prey as a Random Point Process
• Dyson's Brownian-motion model for random matrix theory - revisited. With an Appendix by Don Zagier