On the moments of the characteristic polynomial of a Ginibre random matrix
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Authors
Webb, Christian
Wong, Mo Dick
Publication Date
2019-04-26Journal Title
Proceedings of the London Mathematical Society
ISSN
0024-6115
Publisher
Wiley Online
Volume
118
Issue
5
Pages
1017-1056
Type
Article
This Version
VoR
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Webb, C., & Wong, M. D. (2019). On the moments of the characteristic polynomial of a Ginibre random matrix. Proceedings of the London Mathematical Society, 118 (5), 1017-1056. https://doi.org/10.1112/plms.12225
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, $\Re(\gamma)>-2$, 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.
Sponsorship
Christian Webb was supported by the Academy of Finland grants 288318 and 308123.
Mo Dick Wong is supported by the Croucher Foundation Scholarship and EPSRC grant EP/L016516/1 for his PhD study at Cambridge Centre for Analysis.
Funder references
Engineering and Physical Sciences Research Council (EP/L016516/1)
Identifiers
External DOI: https://doi.org/10.1112/plms.12225
This record's URL: https://www.repository.cam.ac.uk/handle/1810/288455
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