  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.
