Tomas Berggren: Mesoscopic fluctuations for the thinned CUE

I will discus the asymptotic behavior of mesoscopic fluctuations for the thinned Circular Unitary Ensemble. The effect of thinning is that the eigenvalues  start to decorrelate. The decorrelation is  stronger on the larger scales than on the smaller scales.  In the paper ``Mesoscopic fluctuations for the thinned Circular Unitary Ensemble'', by me and Maurice Duits, we investigate this behavior by studying mesoscopic linear statistics. There are two regimes depending on the scale parameter and the thinning parameter.  In one regime we obtain  a CLT of a classical type and in the other regime we retrieve the  CLT for CUE. The two regimes are separated by a critical line. On the critical line the limiting fluctuations are no longer Gaussian, but described by  infinitely divisible laws. 

If there is time, I will mention some ideas of the proofs which are based on a Riemann-Hilbert problem for integrable operators.