On the moments of the characteristic polynomial of a Ginibre random matrix
dc.contributor.author  Webb, Christian  
dc.contributor.author  Wong, Mo Dick  
dc.date.accessioned  20190129T00:30:38Z  
dc.date.available  20190129T00:30:38Z  
dc.date.issued  20190426  
dc.identifier.issn  00246115  
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 $\mathbb{E}\det(G_Nz)^{\gamma}$, 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 FisherHartwig type of singularity: $\det(\int_\mathbb{C} w^{i}\overline{w}^j wz^\gamma e^{Nw^{2}}d^2 w)_{i,j=0}^{N1}$. 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 $wz^\gamma e^{Nw^2}d^2 w$ along with differential identities familiar from the study of asymptotics of Toeplitz and Hankel determinants with FisherHartwig 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 twodimensional set and a FisherHartwig 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  20180628  
rioxxterms.versionofrecord  10.1112/plms.12225  
rioxxterms.version  VoR  
rioxxterms.licenseref.uri  http://creativecommons.org/licenses/by/4.0/  
rioxxterms.licenseref.startdate  20190426  
dc.identifier.eissn  1460244X  
rioxxterms.type  Journal Article/Review  
pubs.funderprojectid  Engineering and Physical Sciences Research Council (EP/L016516/1)  
cam.issuedOnline  20181212  
rioxxterms.freetoread.startdate  20220128 
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