# Limit theorems for statistics of non-crossing partitions

## TL;DR

This paper investigates the statistical properties of large non-crossing partitions, establishing limit theorems and distributional behaviors for various statistics, revealing differences from ordinary set partitions.

## Contribution

It provides new limit theorems and distributional results for statistics of large non-crossing partitions, including Gaussian, geometric, double exponential, and Theta distributions.

## Key findings

- Number of blocks of fixed size follows a Gaussian limit.
- Block sizes are negatively correlated and follow a geometric distribution.
- Largest block size concentrates around log2(n) and follows a double exponential distribution.

## Abstract

We study the distribution of several statistics of large non-crossing partitions. First, we prove the Gaussian limit theorem for the number of blocks of a given fixed size. In contrast to the properties of usual set partitions, we show that the number of blocks of different sizes are negatively correlated, even for large partitions. In addition, we show that the sizes of blocks in a given large non-crossing partition are distributed according to a geometric distribution and not Poisson, as in the case of usual set partitions. Next, we show that the size of the largest block concentrates at $\log_2 n$, and that after an appropriate rescaling, it can be described by the double exponential distribution. Finally, we show that the width of a large non-crossing partition converges to the Theta-distribution which arises in the theory of Brownian excursions.