Counter-ion density profile around a charged disk: from the weak to the strong association regime

We present a comprehensive study of the two dimensional one component plasma in the cell model with charged boundaries. Departing from weak couplings through a convenient approximation of the interacting potential we were able to obtain an analytic formulation to the problem deriving the partition function, density profile, contact densities and integrated profiles that compared well with the numerical data from Monte-Carlo simulations. Additionally, we derived the exact solution for the special cases of $\Xi = 1, 2, 3, ...$ finding a correspondence between the results from weak couplings and exact results. Furthermore, we investigated the strong coupling regime taking into consideration the Wigner formulation. Departing from this, we procured the profile to leading order, computed the contact density values as compared to those derived in a work on the contact theorem. We formulated adequately the strong coupling regime for this system that differed from previous formulations. Ultimately, we computed the first order corrections and compared those results against numerical results from our simulations obtaining very good agreement; this results compared equally well in the planar limit, whose results are known.