Some results on linear statistics for random permutation matrices will be presented, based on joint works with G. Ben Arous (NYU) and D. Zeindler (Bielefeld).
Fluctuations of linear statistics of random matrices have been well studied. Their asymptotic behavior is in general Gaussian. By the work of Diaconis-Shashahani and Diaconis-Evans, matrices from the unitary ensembles follow this behavior. If we consider the subgroup of permutation matrices, a Gaussian limit theorem has been proven by Wieand for the case where the linear statistic counts the number of eigenvalues in a given arc. In the case where the linear statistic is the logarithm of the characteristic polynomial, we obtain a Gaussian limit theorem as well (by results of Hambly-Keevash-O’Connell-Stark, Dang-Zeindler, Zeindler).
But by a recent result of Ben Arous-Dang, the behavior of linear statistics for general functions behave differently, due to the discreteness of the subgroup. In fact, if the functions are smooth enough, the limit law is an infinitely divisible distribution.