The main object of this talk will be a model of $n$ interacting particles at equilibrium. I will describe its macroscopic behavior as $n$ grows to infinity by showing a Laplace principle or, equivalently, a large deviation principle. This implies, in some cases, an almost sure convergence to a deterministic probability measure. Among the main motivating examples we may find Coulomb gases on Riemannian manifolds, the eigenvalue distribution of Gaussian random matrices and the roots of Gaussian random polynomials. This talk is based on arXiv:1703.02680.

The talk is based on joint work with Juan Criado del Rey.
We study a model of a large number of points on the unit sphere under the influence of a finite number of fixed repelling charges. In the large n limit the points fill a region that is known as the droplet. For small external charges the droplet is the complement of the union of a number of spherical caps, one around each of the external charges. When the external charges grow, the spherical caps will start to overlap and the droplet ondergoes a non-trivial deformation.
We explicitly describe the transition for the case of equal external charges that are symmetrically located around the north pole. In our approach we first identify a motherbody that, due to the symmetry in the problem, will be located on a number of meridians connecting the north and south poles. After projecting onto the complex plane, and undoing the symmetry, we characterize the motherbody by means of the solution of a vector equilibrium problem from logarithmic potential theory.