A transition of limiting distributions of large matchings in random graphs

TL;DR

This paper investigates how the distribution of the number of matchings of size b4 in random graphs transitions from normal to log-normal as b4 increases, identifying the critical point of this change.

Contribution

It characterizes the transition point of the limiting distribution of matchings in random graphs for various sizes and probabilities.

Findings

Distribution shifts from normal to log-normal as b4 increases.

Critical b4 value depends on n and p.

Provides a detailed analysis of the transition behavior.

Abstract

We study the asymptotic distribution of the number of matchings of size $\ell=\ell(n)$ in $G(n,p)$ for a wide range of $p=p(n)\in (0,1)$ and for every $1\le \ell\le \lfloor n/2\rfloor$. We prove that this distribution changes from normal to log-normal as $\ell$ increases, and we determine the critical value of $\ell$, as a function of $n$ and $p$, at which the transition of the limiting distribution occurs.