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Random normal matrices and Ward identities

Ameur, Yacin and Hedenmalm, Haakan and Makarov, Nikolai (2015) Random normal matrices and Ward identities. Annals of Probability, 43 (3). pp. 1157-1201. ISSN 0091-1798.

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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.

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Additional Information:© 2015 Institute of Mathematical Statistics. Received February 2013; revised September 2013. Supported by Göran Gustafsson Foundation (KVA). Supported by Göran Gustafsson Foundation (KVA) and by Vetenskapsrådet (VR). Supported by NSF Grant 0201893.
Funding AgencyGrant Number
Göran Gustafsson FoundationUNSPECIFIED
Subject Keywords:Random normal matrix; eigenvalues; Ginibre ensemble; Ward identity; loop equation; Gaussian free field
Issue or Number:3
Classification Code:MSC2010 subject classifications: 60B20, 15B52, 46E22
Record Number:CaltechAUTHORS:20150619-092050374
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:58372
Deposited By: Tony Diaz
Deposited On:19 Jun 2015 16:33
Last Modified:03 Oct 2019 08:35

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