The Annals of Probability

Random normal matrices and Ward identities

Yacin Ameur, Haakan Hedenmalm, and Nikolai Makarov

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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.

Article information

Source
Ann. Probab., Volume 43, Number 3 (2015), 1157-1201.

Dates
First available in Project Euclid: 5 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.aop/1430830280

Digital Object Identifier
doi:10.1214/13-AOP885

Mathematical Reviews number (MathSciNet)
MR3342661

Zentralblatt MATH identifier
06455731

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 15B52: Random matrices 46E22: Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32]

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

Citation

Ameur, Yacin; Hedenmalm, Haakan; Makarov, Nikolai. Random normal matrices and Ward identities. Ann. Probab. 43 (2015), no. 3, 1157--1201. doi:10.1214/13-AOP885. https://projecteuclid.org/euclid.aop/1430830280


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References

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