  We study the asymptotics of certain measures on partitions (the so-called
z-measures and their relatives) in two different regimes: near the diagonal of
the corresponding Young diagram and in the intermediate zone between the
diagonal and the edge of the Young diagram. We prove that in both cases the
limit correlation functions have determinantal form with a correlation kernel
which depends on two real parameters. In the first case the correlation kernel
is discrete, and it has a simple expression in terms of the gamma functions. In
the second case the correlation kernel is continuous and translationally
invariant, and it can be a written as a ratio of two suitably scaled hyperbolic
sines.
