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

Abstract: We study the characteristic polynomial p(n)(x) = Pi(n)(j=1) (|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 E[e(u/pi Im ln pn (r)) e(a Re ln pn (r))], in the case where r is in the bulk, u is an element of R and a is an element of N. This expectation involves an n x 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.

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