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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?

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I think the answer is contained in a more careful reading of the paper above.

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.

On the other hand, the point process describing the smallest Frobenius coordinates requires no such scaling. The implication is then that one expects to young diagram to become ever more elongated (a thinner L shape) as the size increases.

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