On the moments of the characteristic polynomial of a Ginibre random matrix

Abstract

In this article we study the large $N$ asymptotics of complex moments of the absolute value of the characteristic polynomial of a $N\times N$ complex Ginibre random matrix with the characteristic polynomial evaluated at a point in the unit disk. More precisely, we calculate the large $N$ asymptotics of $\mathbb{E}|\det (G_N-z){|}^{\gamma }$, where $G_N$ is a $N\times N$ matrix whose entries are i.i.d and distributed as $N^{-1/2}Z$, $Z$ being a standard complex Gaussian, $\mathrm{Re}(\gamma )>-2$, and $|z|<1$. This expectation is proportional to the determinant of a complex moment matrix with a symbol which is supported in the whole complex plane and has a Fisher-Hartwig type of singularity: $\det ({\int }_{\mathbb{C}}{w}^i{\overline{w}}^j|w-z{|}^{\gamma }{e}^{-N|w{|}^2}{d}^{2}w{)}_{i,j=0}^{N-1}$. We study the asymptotics of this determinant using recent results due to Lee and Yang concerning the asymptotics of orthogonal polynomials with respect to the weight $|w-z{|}^{\gamma }{e}^{-N|w{|}^2}{d}^{2}w$ along with differential identities familiar from the study of asymptotics of Toeplitz and Hankel determinants with Fisher-Hartwig singularities. To our knowledge, even in the case of one singularity, the asymptotics of the determinant of such a moment matrix whose symbol has support in a two-dimensional set and a Fisher-Hartwig singularity, have been previously unknown.

arXiv commentResult and proof extended to include complex-valued gamma; some reference added

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• Strong Asymptotics of Planar Orthogonal Polynomials: Gaussian Weight Perturbed by Finite Number of Point Charges
• Exponential moments for disk counting statistics of random normal matrices in the critical regime
• Planar Orthogonal Polynomials As Type II Multiple Orthogonal Polynomials
• On the Correlation Functions of the Characteristic Polynomials of Non-Hermitian Random Matrices with Independent Entries
• On the critical-subcritical moments of moments of random characteristic polynomials: a GMC perspective
• Orthogonal polynomials in the normal matrix model with two insertions