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

TL;DR

This paper analyzes the asymptotic behavior of moments of the Kerov-Markov-Krein transform of Poissonized Plancharel measures, revealing a recursive structure linked to Catalan numbers and drawing parallels with GUE eigenvalue moments.

Contribution

It establishes a recursive rational expression for moments of the measure's transform in the Poissonized Plancharel case, connecting to Catalan numbers and GUE moment structures.

Findings

Recursive structure of moments expressed as rational functions

Connection between moments and Catalan number generating functions

Analogy with GUE eigenvalue moment structure

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.