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

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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.

This talk is part of the Isaac Newton Institute Seminar Series series.

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