In this article, we consider a fairly general potential in the plane and the corresponding Boltzmann-Gibbs distribution of eigenvalues of random normal matrices. As the order of the matrices tends to infinity, the eigenvalues condensate on a certain compact subset of the plane—the “droplet.” We prove that fluctuations of linear statistics of eigenvalues of random normal matrices converge on compact subsets of the interior of the droplet to a Gaussian field, and we discuss various ramifications of this result.
"Fluctuations of eigenvalues of random normal matrices." Duke Math. J. 159 (1) 31 - 81, 15 July 2011. https://doi.org/10.1215/00127094-1384782