Something old and something new: fixed-point statistics and Deligne-type categories

[Joint work with A. Forey and J. Fresán]

The determination of the limiting Poisson distribution for the number of fixed points of a random permutation, known as the "problème des rencontres", goes back to the early 18th Century, in works of Montmort, N. Bernoulli I and de Moivre. I will present a new approach to this problem, and some of its variants, based on the notion of spectral measure for objects in tensor categories. This method also leads to an interpretation of the Poisson and Gaussian distributions as generalized Sato-Tate measures, and gives rise to interesting arithmetic speculations.