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Random normal matrices and ward identities
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-4971-7147
2015 (English)In: Annals of Probability, ISSN 0091-1798, Vol. 43, no 3, 1157-1201 p.Article in journal (Refereed) Published
Abstract [en]

We consider the random normal matrix ensemble associated with a potential in the plane of sufficient growth near infinity. It is known that asymptotically as the order of the random matrix increases indefinitely, the eigenvalues approach a certain equilibrium density, given in terms of Frostman's solution to the minimum energy problem of weighted logarithmic potential theory. At a finer scale, we may consider fluctuations of eigenvalues about the equilibrium. In the present paper, we give the correction to the expectation of the fluctuations, and we show that the potential field of the corrected fluctuations converge on smooth test functions to a Gaussian free field with free boundary conditions on the droplet associated with the potential.

Place, publisher, year, edition, pages
2015. Vol. 43, no 3, 1157-1201 p.
Keyword [en]
Random normal matrix, eigenvalues, Ginibre ensemble, Ward identity, loop equation, Gaussian free field
National Category
Probability Theory and Statistics
URN: urn:nbn:se:kth:diva-172251DOI: 10.1214/13-AOP885ISI: 000354665200007ScopusID: 2-s2.0-84929252808OAI: diva2:847742
Göran Gustafsson Foundation for Research in Natural Sciences and MedicineSwedish Research Council

QC 20150821

Available from: 2015-08-21 Created: 2015-08-14 Last updated: 2015-08-21Bibliographically approved

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Hedenmalm, Håkan
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