  We consider a macroion confined to a cylindrical cell and neutralized by
oppositely charged counterions. Exact results are obtained for the
two-dimensional version of this problem, in which ion-ion and ion-macroion
interactions are logarithmic. In particular, the threshold for counterion
condensation is found to be the same as predicted by mean-field theory. With
further increase of the macroion charge, a series of single-ion condensation
transitions takes place. Our analytical results are expected to be exact in the
vicinity of these transitions and are in very good agreement with recent
Monte-Carlo simulation data.
