Advanced

Random normal matrices and Ward identities

Ameur, Yacin LU ; Hedenmalm, Haakan and Makarov, Nikolai (2015) In Annals of Probability 43(3). p.1157-1201
Abstract
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.
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Gaussian free field, equation, loop, Ward identity, Ginibre ensemble, eigenvalues, Random normal matrix
in
Annals of Probability
volume
43
issue
3
pages
1157 - 1201
publisher
Institute of Mathematical Statistics
external identifiers
  • wos:000354665200007
  • scopus:84929252808
ISSN
0091-1798
DOI
10.1214/13-AOP885
language
English
LU publication?
yes
id
6ed642d1-71c9-4b9b-a746-59df0bb03ef6 (old id 7410704)
date added to LUP
2015-06-29 11:12:27
date last changed
2017-11-19 03:43:20
@article{6ed642d1-71c9-4b9b-a746-59df0bb03ef6,
  abstract     = {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.},
  author       = {Ameur, Yacin and Hedenmalm, Haakan and Makarov, Nikolai},
  issn         = {0091-1798},
  keyword      = {Gaussian free field,equation,loop,Ward identity,Ginibre ensemble,eigenvalues,Random normal matrix},
  language     = {eng},
  number       = {3},
  pages        = {1157--1201},
  publisher    = {Institute of Mathematical Statistics},
  series       = {Annals of Probability},
  title        = {Random normal matrices and Ward identities},
  url          = {http://dx.doi.org/10.1214/13-AOP885},
  volume       = {43},
  year         = {2015},
}