  We consider a one-dimensional classical Wigner jellium, not necessarily
charge neutral, for which the electrons are allowed to exist beyond the support
of the background charge. The model can be seen as a one-dimensional Coulomb
gas in which the external field is generated by a smeared background on an
interval. It is a true one-dimensional Coulomb gas and not a one-dimensional
log-gas. The system exists if and only if the total background charge is
greater than the number of electrons minus one. For various backgrounds, we
show convergence to point processes, at the edge of the support of the
background. In particular, this provides asymptotic analysis of the
fluctuations of the right-most particle. Our analysis reveals that these
fluctuations are not universal, in the sense that depending on the background,
the tails range anywhere from exponential to Gaussian-like behavior, including
for instance Tracy-Widom-like behavior. We also obtain a Renyi-type
probabilistic representation for the order statistics of the particle system
beyond the support of the background.
