# No limit shape for random Young diagrams under z-measure?

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?

If one is interested in the point process describing the largest Frobenius coordinates of the partition in the limit of large partitions, one has to scale the coordinates by a parameter proportional to the size (it corresponds to the $(1-\xi)$, where $\xi$, is the parameter defining the negative binomial distribution.