The Order of the Giant Component of Random Hypergraphs

TL;DR

This paper proves central and local limit theorems for the size of the largest component in a random hypergraph, using a new probabilistic approach involving Stein's method and multi-round exposure.

Contribution

It introduces a novel probabilistic method for analyzing the giant component in random hypergraphs, extending classical results to hypergraph models.

Findings

Established limit theorems for the largest component size

Developed a new probabilistic proof technique

Applied Stein's method to hypergraph component analysis

Abstract

We establish central and local limit theorems for the number of vertices in the largest component of a random $d$-uniform hypergraph $\hnp$ with edge probability $p=c/\binnd$, where $(d-1)^{-1}+\eps<c<\infty$. 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.