Creation and Growth of Components in a Random Hypergraph Process
Résumé
Denote by an $\ell$-component a connected $b$-uniform hypergraph with $k$ edges and $k(b-1) - \ell$ vertices. We prove that the expected number of creations of $\ell$-component during a random hypergraph process tends to $1$ as $\ell$ and $b$ tend to $\infty$ with the total number of vertices $n$ such that $\ell = o\left( \sqrt[3]{\frac{n}{b}} \right)$. Under the same conditions, we also show that the expected number of vertices that ever belong to an $\ell$-component is approximately $12^{1/3} (b-1)^{1/3} \ell^{1/3} n^{2/3}$. As an immediate consequence, it follows that with high probability the largest $\ell$-component during the process is of size $O( (b-1)^{1/3} \ell^{1/3} n^{2/3} )$. Our results give insight about the size of giant components inside the phase transition of random hypergraphs.
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