Fine structure of moments of the KMK transform of the Poissonized Plancharel measure

Abstract

We consider asymptotics behavior of Poissonized Plancharel measures as the poissonization parameter $N$ goes to infinity. Recently Moll proved a convergent series expansion for statistics of a measure ${\mu }_{\lambda }$ which is the Kerov-Markov-Krein transform of the signed measure on corners a Jack-random partition $\lambda$. The measure ${\mu }_{\lambda }$ is of interest because it behaves in some ways like the empirical measure on eigenvalues of a GUE-random matrix. We prove for the Poissonized Plancharel case that the large $N$ series for moments of ${\mu }_{\lambda }$ have a recursive structure as rational expressions in the generating function for Catalan numbers. We discuss the analogy between our result and the fine structure of moments of the GUE.