Correlation kernels arising in harmonic analysis on
the infinite symmetric group and the infinite unitary group
Decomposition of natural representations of ``big'' groups like the infinite symmetric group or the infinite unitary group into irreducibles provides a rich source of stochastic point processes with determinantal correlation functions. Correlation kernels are always expressed through the classical special functions and form a hierarchy which is parallel to that of the theory of special functions. The new models have a remarkable resemblance to those of the Random Matrix Theory. The correlation kernels turn out to be more general than the kernels from the Random Matrix Theory. Asymptotic analysis of the kernels leads to important applications in combinatorics, for example, to a proof of the Baik--Deift--Johansson conjecture. This is a joint work with Andrei Okounkov and Grigori Olshanski.