Variance of Linear Statistic for Plancherel Young Diagrams

TL;DR

This paper calculates the exact asymptotics of variance for linear statistics on Plancherel Young diagrams, demonstrating a local configuration with linearly growing variance and establishing a central limit theorem for it.

Contribution

It provides the first precise asymptotic analysis of variance for linear statistics on Plancherel Young diagrams and introduces a new local configuration with linearly growing variance.

Findings

Exact asymptotics of variance for linear statistics on Young diagrams

Identification of a local configuration with linearly growing variance

Proof of the central limit theorem for the identified configuration

Abstract

In this paper we compute the precise asymptotics of the variance of linear statistic of descents on a growing interval for Plancherel Young diagrams (following Vershik and Kerov, diagrams are considered rotated by $\pi/4$). We also give an example of a local configuration with linearly growing variance in a fixed regime and prove the central limit theorem for this configuration in the given regime.