A variational characterization of the sine_\beta point process

The Sineβ point process is a stationary point process that appears as the limit of a certain system of interacting particles with a logarithmic repulsion, the so called one-dimensional log-gas. In this talk, we give a characterisation of the Sineβ process as the unique minimizer of a free energy functional. Our argument is based on the combination of optimal transport ideas for point processes and approximation techniques for log-gases (screening). (Joint work with Matthias Erbar and Thomas Leblé.)