On the characteristic polynomial of the eigenvalue moduli of random normal matrices

2022/05/09 by Byun, Sung-Soo, Charlier, Christophe · 1 citation
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2205.04298

Abstract

We study the characteristic polynomial p_n(x)=∏_j=1ⁿ(|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^\fracuπ Im ln p_n(r)e^a Re ln p_n(r)], in the case where r is in the bulk, u ∈ ℝ and a ∈ ℕ. This expectation involves an 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.

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