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Random normal matrices and Ward identities

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Author

Summary, in English

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.

Publishing year

2015

Language

English

Pages

1157-1201

Publication/Series

Annals of Probability

Volume

43

Issue

3

Document type

Journal article

Publisher

Institute of Mathematical Statistics

Topic

  • Probability Theory and Statistics

Keywords

  • Gaussian free field
  • equation
  • loop
  • Ward identity
  • Ginibre ensemble
  • eigenvalues
  • Random normal matrix

Status

Published

ISBN/ISSN/Other

  • ISSN: 0091-1798