The double Dyson index β effect in non-Hermitian tridiagonal matrices (original) (raw)

The Dyson index β plays an essential role in random matrix theory as it labels the so-called "three fold way" that refers to the symmetries satisfied by the ensembles under unitary transformations. As it is known, its 1, 2 and 4 values denote the Orthogonal, the Unitary and the Symplectic classes whose matrix elements are real, complex and quaternion numbers, respectively. It functions therefore as a measure of the number of independent non-diagonal variables. On the other hand, in the case of the β-ensembles, which are the tridiagonal form of the theory, it can assume any real positive value loosing this way that function. Our purpose, however is to show that, when the Hermitian condition of the real matrices generated with a given value of β, is removed, and, as a consequence, the number of non-diagonal independent variables doubles, non-Hermitian matrices exist that asymptotically behave as if they had been generated with a value 2β. So, it is as if the β index were, in this way, again operative. It is shown that this effect happens for the three tridiagonal ensembles, namely, the β-Hermite, the β-Laguerre and the β-Jacobi ensemble.