Random normal matrices and Ward identities

TL;DR

This paper studies the fluctuations of eigenvalues in random normal matrices, providing corrections to their expected behavior and proving convergence to a Gaussian free field under certain conditions.

Contribution

It introduces a correction to the expected eigenvalue fluctuations and establishes their convergence to a Gaussian free field with free boundary conditions.

Findings

Correction to the expectation of eigenvalue fluctuations

Convergence of the potential field to a Gaussian free field

Results applicable to ensembles with strong potentials near infinity

Abstract

Consider the random normal matrix ensemble associated with a potential on the plane which is sufficiently strong near infinity. It is known that, to a first approximation, the eigenvalues obey a certain equilibrium distribution, given by Frostman's solution to the minimum energy problem of weighted logarithmic potential theory. On a finer scale, one can consider fluctuations of eigenvalues about the equilibrium. In the present paper, we give the correction to the expectation of fluctuations, and we prove 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.