Counting Points in Boxes: the Riesz Family & Friends

Members of the Riesz family are statistical physics systems that include Coulomb and log-gases. Each one gives rise to a (Gibbsian) point process, and a basic question is to study the random number of points in large d-dimensional boxes. In some "solvable" cases, one finds interesting properties like hyperuniformity and (number)-rigidity as well as a tripartite "large deviations" estimate. "Universality" of such features with respect to the interaction, the temperature, the external field is mostly open. I will present several questions (and few results) around this theme.

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