Asymptotics of q-Plancherel measures

TL;DR

This paper investigates the asymptotic behavior of the first rows and columns of random Young diagrams under q-Plancherel measures, revealing explicit first- and second-order asymptotics using polynomial functions on Young diagrams.

Contribution

It provides explicit asymptotic formulas for the largest rows of Young diagrams under q-Plancherel measures, extending previous results to new measures from Hecke algebras.

Findings

First rows are typically of order n.

Explicit first- and second-order asymptotics are computed.

Method applies to other measures like Schur-Weyl representations.

Abstract

In this paper, we are interested in the asymptotic size of rows and columns of a random Young diagram under a natural deformation of the Plancherel measure coming from Hecke algebras. The first lines of such diagrams are typically of order $n$, so it does not fit in the context studied by P. Biane and P. \'Sniady. Using the theory of polynomial functions on Young diagrams of Kerov and Olshanski, we are able to compute explicitly the first- and second-order asymptotics of the length of the first rows. Our method works also for other measures, for instance those coming from Schur-Weyl representations.