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

Abstract

AbstractWe study the characteristic polynomial $$p_{n}(x)=\prod _{j=1}^{n}(|z_{j}|-x)$$ p n ( x ) = ∏ j = 1 n ( | z j | - x ) where the $$z_{j}$$ 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 $$\mathbb {E}[e^{\frac{u}{\pi } \, \text {Im\,}\ln p_{n}(r)}e^{a \, \text {Re\,}\ln p_{n}(r)}]$$ E [ e u π Im\, ln p n ( r ) e a Re\, ln p n ( r ) ] , in the case where r is in the bulk, $$u \in \mathbb {R}$$ u ∈ R and $$a \in \mathbb {N}$$ a ∈ N . This expectation involves an $$n \times n$$ n × 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.