  We study equilibrium statistical mechanics of classical point counter-ions,
formulated on 2D Euclidean space with logarithmic Coulomb interactions
(infinite number of particles) or on the cylinder surface (finite particle
numbers), in the vicinity of a single uniformly charged line (one single
double-layer), or between two such lines (interacting double-layers). The
weak-coupling Poisson-Boltzmann theory, which applies when the coupling
constant Gamma is small, is briefly recapitulated (the coupling constant is
defined as Gamma = beta e^2 where beta is the inverse temperature, and e the
counter-ion charge). The opposite strong-coupling limit (Gamma -> infinity) is
treated by using a recent method based on an exact expansion around the
ground-state Wigner crystal of counter-ions. The weak- and strong-coupling
theories are compared at intermediary values of the coupling constant Gamma=2
gamma (gamma=1,2,3), to exact results derived within a 1D lattice
representation of 2D Coulomb systems in terms of anti-commuting field
variables. The models (density profile, pressure) are solved exactly for any
particles numbers N at Gamma=2 and up to relatively large finite N at Gamma=4
and 6. For the one-line geometry, the decay of the density profile at
asymptotic distance from the line undergoes a fundamental change with respect
to the mean-field behavior at Gamma=6. The like-charge attraction regime,
possible in the strong coupling limit but precluded at mean-field level,
survives for Gamma=4 and 6, but disappears at Gamma=2.
