Large-N asymptotics of energy-minimizing measures on N-point configurations

If N points interact by Coulomb 2-point repulsion and under a "confining" potential V(x)=|x|^2, as N goes to infinity they spread uniformly in a ball. This is a typical problem about "energy-minimizing configurations".  What is the simplest problem that we get if we move from variational problems on N-point configurations, to variational problems on measures on N-point configurations? In that case there is a more natural replacement of the "confinement", previously played by V(x): it is to just "fix the 1-point marginal" of our measure on configurations. We obtain a generalization of optimal transport, for N-marginals instead of the usual 2-marginals case.   In my talk I'll describe the above two types of large-N asymptotics problems in more detail, I'll overview the techniques that we know, and I'll mention some parts of this subject that we currently don't understand.