Let $H_k(n,p)$ be the random $k$-uniform hypergraph on $n$ vertices, where each hyperedge $E\subseteq\{1,\dots,n\}$, $|E|=k$, is included independently with probability $p$. Further, let $d=\binom{n-1}{k-1}p$ be the average vertex degree. For a long time, we know (e.g. from here or here) that a large component emerges at $\alpha=\frac{1}{k-1}$, meaning that for $\alpha\in(0,\frac{1}{k-1})$ and $p=\alpha/\binom{n-1}{k-1}$ the size of the largest component is $\mathcal O(\ln(n))$, while for $\alpha>\frac{1}{k-1}$ it is $\Theta(n)$. For $\alpha<\frac{1}{k-1}$ the hypergraph is subcritical, and supercritical for $\alpha>\frac{1}{k-1}$.
For graphs, i.e. the special case $k=2$, we have a thorough understanding of the subcritical (and supercritical) regime, as discussed here, here, here and in countless other books. In particular, say Corollary 5.8 in this book, we know that in the subcritical regime all connected components are trees or unicyclic, and the number of unicyclic components is at most $f(n)$ for any $f(n)=\omega(1)$, with probability tending to $1$.
Now, we turn to hypergraphs. A connected hypergraph on $n$ vertices with $m$ hyperdges is a hypertree if $n=(k-1)m+1$ (isolated vertices are hypertrees), and unicyclic if $n=(k-1)m$.
Questions
- Does Corollary 5.8 hold for all $k\ge 2$, i.e. is it true that for $\alpha<\frac{1}{k-1}$ all connected components are trees or unicyclic, and the number of unicyclic components is at most $f(n)$ for any $f(n)=\omega(1)$, with probability tending to $1$?
- If not, does there exist such a characterization of the subcritical regime for $k>2$?
- Does there exist any book on random (binomial or uniform) hypergraphs covering the standard results?
The only related result I found so far is in Section 9.4.1 in the blue book (for the uniform model on labeled hyperedges), where it's established that the expected ratio of vertices covered by finite size trees approaches $1$, which suggests that the ratio of vertices covered by trees approaches $1$, with probability tending to $1$.