dc.contributor.author Webb, Christian dc.contributor.author Wong, Mo Dick dc.date.accessioned 2019-01-29T00:30:38Z dc.date.available 2019-01-29T00:30:38Z dc.date.issued 2019-04-26 dc.identifier.issn 0024-6115 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/288455 dc.description.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. dc.description.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. dc.publisher Wiley Online dc.rights Attribution 4.0 International dc.rights.uri http://creativecommons.org/licenses/by/4.0/ dc.title On the moments of the characteristic polynomial of a Ginibre random matrix dc.type Article prism.endingPage 1056 prism.issueIdentifier 5 prism.publicationDate 2019 prism.publicationName Proceedings of the London Mathematical Society prism.startingPage 1017 prism.volume 118 dc.identifier.doi 10.17863/CAM.35743 dcterms.dateAccepted 2018-06-28 rioxxterms.versionofrecord 10.1112/plms.12225 rioxxterms.version VoR rioxxterms.licenseref.uri http://creativecommons.org/licenses/by/4.0/ rioxxterms.licenseref.startdate 2019-04-26 dc.identifier.eissn 1460-244X rioxxterms.type Journal Article/Review pubs.funder-project-id Engineering and Physical Sciences Research Council (EP/L016516/1) cam.issuedOnline 2018-12-12 rioxxterms.freetoread.startdate 2022-01-28
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