  We investigate the electrostatic equilibria of N discrete charges of size 1/N
on a two dimensional conductor (domain). We study the distribution of the
charges on symmetric domains including the ellipse, the hypotrochoid and
various regular polygons, with an emphasis on understanding the distributions
of the charges, as the shape of the underlying conductor becomes singular. We
find that there are two regimes of behavior, a symmetric regime for smooth
conductors, and a symmetry broken regime for ``singular'' domains. For smooth
conductors, the locations of the charges can be determined up to a certain
order by an integral equation due to Pommerenke (1969). We present a derivation
of a related (but different) integral equation, which has the same solutions.
We also solve the equation to obtain (asymptotic) solutions which show
universal behavior in the distribution of the charges in conductors with
somewhat smooth cusps. Conductors with sharp cusps and singularities show
qualitatively different behavior, where the symmetry of the problem is broken,
and the distribution of the discrete charges does not respect the symmetry of
the underlying domain. We investigate the symmetry breaking both theoretically,
and numerically, and find good agreement between our theory and the numerics.
We also find that the universality in the distribution of the charges near the
cusps persists in the symmetry broken regime, although this distribution is
very different from the one given by the integral equation.
