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?