A transition of limiting distributions of large matchings in random graphs

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