The genus of a random chord diagram is asymptotically normal

TL;DR

This paper proves that the genus of a random chord diagram, formed by gluing sides of an n-gon, follows an asymptotically normal distribution with specific mean and variance, extending previous expected value results.

Contribution

It establishes a local limit theorem showing the genus distribution is asymptotically Gaussian, providing detailed probabilistic behavior beyond the expected value.

Findings

Genus distribution is asymptotically normal.

Mean of genus is approximately (n - ln n)/2.

Variance of genus is approximately (ln n)/4.

Abstract

Let $G_n$ be the genus of a two-dimensional surface obtained by gluing, uniformly at random, the sides of an $n$-gon. Recently Linial and Nowik proved, via an enumerational formula due to Harer and Zagier, that the expected value of $G_n$ is asymptotic to $(n - \ln n)/2$ for $n\to\infty$. We prove a local limit theorem for the distribution of $G_n$, which implies that $G_n$ is asymptotically Gaussian, with mean $(n-\ln n)/2$ and variance $(\ln n)/4$.