We consider two related problems: the first is the minimization of the "Coulomb renormalized energy" of Sandier-Serfaty, which corresponds to the total Coulomb interaction of point charges in a uniform neutralizing background (or rather variants of it). The second corresponds to the minimization of the Hamiltonian of a two-dimensional "Coulomb gas" or "one-component plasma", a system of n point charges with Coulomb pair interaction, in a confining potential (minimizers of this energy also correspond to "weighted Fekete sets"). In both cases we investigate the microscopic structure of minimizers, i.e. at the scale corresponding to the interparticle distance. We show that in any large enough microscopic set, the value of the energy and the number of points are "rigid" and completely determined by the macroscopic density of points. In other words, points and energy are "equidistributed" in space (modulo appropriate scalings). The number of points in a ball is in particular known up to an error proportional to the radius of the ball. We also prove a result on the maximal and minimal distances between points. Our approach involves fully exploiting the minimality by reducing to minimization problems with fixed boundary conditions posed on smaller subsets.
