On the moments of the characteristic polynomial of a Ginibre random matrix
| cam.issuedOnline | 2018-12-12 | |
| 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.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.identifier.doi | 10.17863/CAM.35743 | |
| dc.identifier.eissn | 1460-244X | |
| dc.identifier.issn | 0024-6115 | |
| dc.identifier.uri | https://www.repository.cam.ac.uk/handle/1810/288455 | |
| dc.language.iso | eng | |
| dc.publisher | Wiley Online | |
| dc.publisher.url | https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/plms.12225 | |
| dc.rights | Attribution 4.0 International | |
| dc.rights.uri | http://creativecommons.org/licenses/by/4.0/ | |
| dc.subject | 4901 Applied Mathematics | |
| dc.subject | 4902 Mathematical Physics | |
| dc.subject | 49 Mathematical Sciences | |
| dc.title | On the moments of the characteristic polynomial of a Ginibre random matrix | |
| dc.type | Article | |
| dcterms.dateAccepted | 2018-06-28 | |
| prism.endingPage | 1056 | |
| prism.issueIdentifier | 5 | |
| prism.publicationDate | 2019 | |
| prism.publicationName | Proceedings of the London Mathematical Society | |
| prism.startingPage | 1017 | |
| prism.volume | 118 | |
| pubs.funder-project-id | Engineering and Physical Sciences Research Council (EP/L016516/1) | |
| rioxxterms.licenseref.startdate | 2019-04-26 | |
| rioxxterms.licenseref.uri | http://creativecommons.org/licenses/by/4.0/ | |
| rioxxterms.type | Journal Article/Review | |
| rioxxterms.version | VoR | |
| rioxxterms.versionofrecord | 10.1112/plms.12225 |
Files
Original bundle
1 - 1 of 1
Loading...
- Name:
- plms.12225.pdf
- Size:
- 604.05 KB
- Format:
- Adobe Portable Document Format
- Description:
- Published version
- Licence
- http://creativecommons.org/licenses/by/4.0/
License bundle
1 - 1 of 1
No Thumbnail Available
- Name:
- DepositLicenceAgreementv2.1.pdf
- Size:
- 150.9 KB
- Format:
- Adobe Portable Document Format