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

We study the characteristic polynomial pₙ(x)=∏ ⱼ₌₁ⁿ(|zⱼ|-x) p n ( x ) = ∏ j = 1 n ( | z j | - x ) where the zⱼ 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 E[e^u/π \, Im\,ln pₙ(r)e^a \, Re\,ln pₙ(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 ∈ R u ∈ R and a ∈ N a ∈ N . This expectation involves an n × 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 .