  We discuss the 1/N expansion of the free energy of N logarithmically
interacting charges in the plane in an external field. For some particular
values of the inverse temperature beta this system is equivalent to the
eigenvalue version of certain random matrix models, where it is refered to as
the "Dyson gas" of eigenvalues. To find the free energy at large N and the
structure of 1/N-corrections, we first use the effective action approach and
then confirm the results by solving the loop equation. The results obtained
give some new representations of the mathematical objects related to the
Dirichlet boundary value problem, complex analysis and spectral geometry of
exterior domains. They also suggest interesting links with bosonic field theory
on Riemann surfaces, gravitational anomalies and topological field theories.
