Integer partitions and exclusion statistics: Limit shapes and the largest part of Young diagrams

TL;DR

This paper analyzes the asymptotic shapes of Young diagrams for minimal difference partitions, revealing a universal Gumbel distribution for the largest part across various partition types.

Contribution

It provides explicit limit shapes for minimal difference partitions and demonstrates the universal Gumbel distribution for the largest part in these and related partitions.

Findings

Limit shapes of Young diagrams for minimal difference partitions are derived.

The scaled distribution of the largest part follows a Gumbel distribution.

The Gumbel distribution remains valid for a broader class of partitions.

Abstract

We compute the limit shapes of the Young diagrams of the minimal difference $p$ partitions and provide a simple physical interpretation for the limit shapes. We also calculate the asymptotic distribution of the largest part of the Young diagram and show that the scaled distribution has a Gumbel form for all $p$. This Gumbel statistics for the largest part remains unchanged even for general partitions of the form $E=\sum_i n_i i^{1/\nu}$ with $\nu>0$ where $n_i$ is the number of times the part $i$ appears.