  In this paper we study 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. 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. We argue that this transition phenomenon is universal by showing that the
same transition and their laws appear for fluctuations of the thinned sine
process in a growing box. The proofs are based on a Riemann-Hilbert problem for
integrable operators.
