  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 \cite{DK}. We
strongly believe that also in other concrete cases the double integral formulas
are good starting points for asymptotic studies.
