Area Statistics for Large Oscillating Tableaux

TL;DR

This paper models the area of partitions in large oscillating tableaux as a weighted random walk in the first quadrant, deriving recursive moments and demonstrating convergence to a Gaussian process as size increases.

Contribution

It introduces a novel probabilistic model for oscillating tableaux and provides recursive formulas for moments, showing asymptotic Gaussian behavior.

Findings

The area distribution is described by a weighted random walk.

Recursive formulas for moments are derived.

Convergence to Gaussian process is proven for large tableaux.

Abstract

In this note we show that the area of the partitions making up an oscillating tableaux is described by a random walk on the first quadrant of $\mathbb{Z}^2$ with certain position dependent weights. We are able to recursively calculate the moments of the walk. As the length of the oscillating tableaux becomes large we show that this random walk converges to a Gaussian stochastic process.