Asymptotics of Some Plancherel Averages via Polynomiality Results

TL;DR

This paper investigates the asymptotic behavior of certain functions of Young diagrams under the Plancherel measure, using polynomiality results to derive explicit formulas and analyze their properties as the diagram size grows.

Contribution

It introduces a novel approach to analyze Plancherel averages by expressing them as binomial convolutions and applying integral and generating function techniques.

Findings

Explicit expression for the constant in the almost equipartition property

Representation of expectations as binomial convolutions

Application of Rice's integral and Poisson generating functions

Abstract

Consider Young diagrams of $n$ boxes distributed according to the Plancherel measure. So those diagrams could be the output of the RSK algorithm, when applied to random permutations of the set $\{1,\ldots,n\}$. Here we are interested in asymptotics, as $n\to \infty$, of expectations of certain functions of random Young diagrams, such as the number of bumping steps of the RSK algorithm that leads to that diagram, the side length of its Durfee square, or the logarithm of its probability. We can express these functions in terms of hook lengths or contents of the boxes of the diagram, which opens the door for application of known polynomiality results for Plancherel averages. We thus obtain representations of expectations as binomial convolutions, that can be further analyzed with the help of Rice's integral or Poisson generating functions. Among our results is a very explicit expression for the constant appearing in the almost equipartition property of the Plancherel measure.