In their paper *Random partitions and the Gamma kernel* (Advances in Mathematics 194 (2005) 141–202), Borodin and Olshanski state that:

An important difference between the Plancherel measures and the z-measures is that the random Plancherel diagrams have a limit form... while no such form exists for the z-measures.

I am not sure if this is a straightforward comment (because the z-measure is in general not positive), or more subtle. That is, **is there still no limit shape for those values where the z-measure is positive?**. If not, why?