The Order of the Giant Component of Random Hypergraphs

We establish central and local limit theorems for the number of vertices in the largest component of a random $d$-uniform hypergraph $\text{hnp}$ with edge probability $p=c\mathrm{/}\text{binnd}$, where $(d-1{)}^{-1}+\text{eps}<c<\mathrm{\infty }$. The proof relies on a new, purely probabilistic approach, and is based on Stein’s method as well as exposing the edges of $H_d(n,p)$ in several rounds.