A matrix model of a non-Hermitian β-ensemble

We introduce the first random matrix model of a complex [Formula: see text]-ensemble. The matrices are tridiagonal and can be thought of as the non-Hermitian analogue of the Hermite [Formula: see text]-ensembles discovered by [I. Dumitriu and A. Edelman, Matrix models for beta ensembles, J. Math. Phys. 43 (2002) 5830–5847]. The main feature of the model is that the exponent [Formula: see text] of the Vandermonde determinant in the joint probability density function (j.p.d.f.) of the eigenvalues can take any value in [Formula: see text] However, when [Formula: see text], the j.p.d.f. does not reduce to that of the Ginibre ensemble, but it contains an extra factor expressed as a multidimensional integral over the space of the eigenvectors.