On the diameter of random uniform hypergraphs in dense regime

Abstract

For a fixed natural number $t\geq 2$, we consider $t$-uniform random hypergraphs $\mathscr{H} (n,t,p)$ on $n$ vertices $[n]=\{1,\ldots, n\}$, where each $t$-subset of $[n]$ is included as a hyperedge with probability $p$ and independently. We show that the diameter of $\mathscr{H} (n,t,p)$ is concentrated only at two points in the dense regime. More precisely, suppose $diam(\mathcal H)$ denotes the diameter of a hypergraph $\mathcal H$ on $n$ vertices. We show that, for fixed $t,c,d$ constants, if $n$ and $p$ (depends on $t,c,d,n$) satisfy $$ \frac{ (t-1)^ {d} N^{d} p^{d}} {n}= \log \left( \frac{n^2}{c} \right), \mbox{ where } N={n-1\choose t-1}, $$ $c$ is a positive constant and $d\geq2$ is a natural number, then $$ \lim_{n \to \infty} \mathbb{P} \left( diam( \mathcal{H}) = d \right) = e^{- \frac{c}{2}} \text{ and } \lim_{n \to \infty} \mathbb{P} \left( diam(\mathcal{H}) = d+1 \right) = 1- e^{- \frac{c}{2}}. $$ In particular, the case where $t = 2$ corresponds to the diameter of the Erd\H{o}s-R\'enyi graph, as established by Bollob\'as in \cite[Theorem~6]{bollobas1981diameter}. Bollob\' as's result was proven using the moments method, which is challenging to apply in our context due to the complexity of the model. In this paper, we utilize the Stein-Chen method along with coupling techniques to prove our result. This approach can potentially be used to solve various problems, in particular diameter problems, in more complex networks.