2D coulomb gas, abrikosov lattice and renormalized energy

In joint work with Etienne Sandier, we studied the statistical mechanics of a classical twodimensional Coulomb gas, particular cases of which also correspond to random matrix ensembles. We connect the problem to the “renormalized energy” W, a Coulombian interaction for an infinite set of points in the plane that we introduced in connection to the Ginzburg-Landau model, and whose minimum is expected to be achieved by the “Abrikosov” triangular lattice. Results include a next order asymptotic expansion of the partition function, and various characterizations of the behavior of the system at the microscopic scale. When the temperature tends to zero we show that the system tends to “crystallize” to a minimizer of W.