Two-dimensional Coulomb Systems in a disk with Ideal Dielectric Boundaries

We consider two-dimensional Coulomb systems confined in a disk with ideal dielectric boundaries. In particular we study the two-component plasma in detail. When the coulombic coupling constant $\Gamma=2$ the model is exactly solvable. We compute the grand-potential, densities and correlations. We show that the grand-potential has a universal logarithmic finite-size correction as predicted in previous works. This logarithmic finite-size correction is also checked in another solvable model: the one-component plasma.