We study the correlation functions for determinantal point processes defined by products of infinite minors of block Toeplitz matrices. The motivation for studying such processes comes from doubly periodically weighted tilings of planar domains, such as the two-periodic Aztec diamond. Our main results are double integral formulas for the correlation kernels. In general, the integrand is a matrix-valued function built out of a factorization of the matrix-valued weight. In concrete examples the factorization can be worked out in detail and we obtain explicit integrands. In particular, we find an alternative proof for a formula for the two-periodic Aztec diamond recently derived in [20]. We strongly believe that also in other concrete cases the double integral formulas are good starting points for asymptotic studies.for a determinantal point process can be expressed in terms of determinants of matrices constructed out of a function of two variables, called the correlation kernel. For Schur processes, this correlation kernel can be written in terms of double integral formulas with explicitly known integrands. Saddle point methods are thus at our disposal for the asymptotic analysis of concrete examples. This has been a large industry in recent years and we do not attempt to provide a full list of works, but point to [1,6,32,33] and the references therein, for introductions to this subject and as general references.An important motivation for studying the extension to block Toeplitz minors comes from random tilings of planar domains or random dimer configurations with doubly-periodic weights, that was discussed in [34,35,40]. The double periodicity in the weight structure naturally leads in many cases to taking minors of block Toeplitz matrices. Being a natural and non-trivial extension of the scalar case one may therefore expect a richer structure where new phenomena can be discovered. A remarkable feature for periodically weighted dimer models is that a so-called gas region [35,40] may appear. In such a region the 2-point correlations for the height function decay exponentially with the distance. However, the integrable structure for these models is relatively unexplored and one reason for this is that the standard techniques for the scalar case are inadequate. Explicit double integrals for the kernel, even for n → ∞, are not known generally. In fact, to the best of our knowledge, such a double integral formula is only known in case of the two-periodic Aztec diamond. The first results are by Chhita and Young [13] and Chhita and Johansson [11], who found a machinery for computing the inverse Kasteleyn matrix explicitly and used that to perform an asymptotic analysis. See also [3] for further results.Of special importance to us is the recent paper [20], where one of us together with Kuijlaars took a different approach for the two-periodic Aztec diamond. Starting from the definition of non-intersecting path ensembles with general block Toeplitz transitions we showed that the correlation kernel can be related to matrix ...

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.

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