On the Characteristic Polynomial of the Eigenvalue Moduli of Random Normal Matrices

We study the characteristic polynomial \(p_{n}(x)=\prod _{j=1}^{n}(|z_{j}|-x)\) where the \(z_{j}\) are drawn from the Mittag–Leffler ensemble, i.e. a two-dimensional determinantal point process which generalizes the Ginibre point process. We obtain precise large n asymptotics for the moment generating function \(\mathbb {E}[e^{\frac{u}{\pi } \, \text {Im\,}\ln p_{n}(r)}e^{a \, \text {Re\,}\ln p_{n}(r)}]\), in the case where r is in the bulk, \(u \in \mathbb {R}\) and \(a \in \mathbb {N}\). This expectation involves an \(n \times n\) determinant whose weight is supported on the whole complex plane, is rotation-invariant, and has both jump- and root-type singularities along the circle centered at 0 of radius r. This “circular" root-type singularity differs from earlier works on Fisher–Hartwig singularities, and surprisingly yields a new kind of ingredient in the asymptotics, the so-called associated Hermite polynomials.