  We investigate non-perturbative corrections to the large $N$ expansion of the
homogeneous electron gas. These are associated with instanton solutions to the
effective action of the plasmon field. We show that, although the large field
behavior of that action dominates the quadratic bare Coulomb term, there are no
solutions at large field, and consequently none at large density. We argue that
solutions would exist at low density if the large $N$ theory had a Wigner
crystal (WC) phase. However, we argue that this is not the case. Together with
the implied convergence of the large $N$ expansion, this implies that the
homogeneous electron gas with $N$ component spins and a Coulomb interaction
scaling like $1/N$ can only have a WC phase below a curve in the plane of $N$
and density, which asymptotes to zero density at infinite $N$. We argue that
for systems with a semi-classical expansion for order parameter dynamics, and a
first order quantum transition between fluid and crystal phases, there are
instantons associated with the decays of meta-stable fluid and crystal phases
in the appropriate regions of the phase diagram. We argue that the crystal will
decay into one or more colloidal or bubble phases\cite{kivspiv} rather than
directly into the fluid. The transition to a translationally invariant phase is
likely to be second order. Unfortunately, the HEG does not have a crystal phase
at large $N$, where these semi-classical ideas could be examined in detail. We
suggest that the evidence for negative dielectric function at intermediate
densities for $N = 2$ is an indicator of this second order transition. It is
possible that the closed large $N$ equation for the plasmon two point function,
derived in\cite{ergheg} might capture at least the qualitative features of the
second order transition.
