Note on a partition limit theorem for rank and crank

TL;DR

This paper characterizes the limiting distribution of the rank and crank statistics of partitions, showing they converge to a distribution involving Brownian motion and extreme value distributions, extending understanding of partition statistics.

Contribution

It identifies the precise limiting distribution of rank and crank, connecting them to Brownian motion and extreme value theory, which was previously unknown.

Findings

Limit distribution is the difference of two independent extreme value distributions.

The same limit applies to both rank and crank statistics.

Limit distribution involves Brownian motion hitting the unit sphere.

Abstract

If L is a partition of n, the rank of L is the size of the largest part minus the number of parts. Under the uniform distribution on partitions, Bringmann, Mahlburg, and Rhoades showed that the rank statistic has a limiting distribution. We identify the limit as the difference between two independent extreme value distributions and as the distribution of B(T) where B(t) is standard Brownian motion and T is the first time that an independent three-dimensional Brownian motion hits the unit sphere. The same limit holds for the crank.