A transition of limiting distributions of large matchings in random graphs

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.