Universal Finite-Size Effects in the Two-Dimensional Asymmetric Coulomb Gas on a Sphere

We consider an asymmetric version of a two-dimensional Coulomb gas, made up of two species of pointlike particles with positive $+1$ and negative -1/Q $(Q = 1, 2, ...)$ charges; Q=1 corresponds to the symmetric two-component plasma and the limiting case $Q\to\infty$ is related to the one-component plasma. The system lives on the surface of a sphere, and it is studied in both canonical and grand-canonical ensembles. By combining the method of stereographic projection of the sphere onto an infinite plane with the technique of a renormalized Mayer series expansion it is explicitly shown that the finite-size expansions of the free energy and of the grand potential have the same universal term, independent of model's details. As a by-product, the collapse temperature and the Kosterlitz-Thouless transition point (in the limit of a vanishing hard-core attached to particles) are conjectured for any value of $Q$.