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    Creation and Growth of Components in a Random Hypergraph Process

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    Denote by an ℓ\ell-component a connected bb-uniform hypergraph with kk edges and k(b−1)−ℓk(b-1) - \ell vertices. We prove that the expected number of creations of ℓ\ell-component during a random hypergraph process tends to 1 as ℓ\ell and bb tend to ∞\infty with the total number of vertices nn such that ℓ=o(nb3)\ell = o(\sqrt[3]{\frac{n}{b}}). Under the same conditions, we also show that the expected number of vertices that ever belong to an ℓ\ell-component is approximately 121/3(b−1)1/3ℓ1/3n2/312^{1/3} (b-1)^{1/3} \ell^{1/3} n^{2/3}. As an immediate consequence, it follows that with high probability the largest ℓ\ell-component during the process is of size O((b−1)1/3ℓ1/3n2/3)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.Comment: R\'{e}sum\'{e} \'{e}tend
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