In this paper, we study the high-order phase transition in random $r$-uniform hypergraphs. For a positive integer $n$ and a real $p\in [0,1]$, let $H:=H^r(n,p)$ be the random $r$-uniform hypergraph with vertex set $[n]$, where each $r$-set is selected as an edge with probability $p$ independently randomly. For $1\leq s \leq r-1$ and two $s$-sets $S$ and $S'$, we say $S$ is connected to $S'$ if there is a sequence of alternating $s$-sets and edges $S_0,F_1,S_1,F_2, \ldots, F_k, S_k$ such that $S_0,S_1,\ldots, S_k$ are $s$-sets, $S_0=S$, $S_k=S'$, $F_1,F_2,\ldots, F_k$ are edges of $H$, and $S_{ i-1}\cup S_i\subseteq F_i$ for each $1\leq i\leq k$. This is an equivalence relation over the family of all $s$-sets ${ [n]\choose s}$ and results in a partition: ${ V\choose s}=\cup_i C_i$. Each $C_i$ is called an { $s$-th-order} connected component and a component $C_i$ is { \em giant} if $|C_i|=\Theta(n^s)$. We prove that the sharp threshold of the existence of the $s$-th-order giant connected components in $H^r(n,p)$ is $\frac{ 1}{ \big({ r\choose s}-1\big){ n\choose r-s}}$. Let $c={ n\choose r-s}p$. If $c$ is a constant and $c<\tfrac{ 1}{ \binom{ r}{ s}-1}$, then with high probability, all $s$-th-order connected components have size $O(\ln n)$. If $c$ is a constant and $c > \tfrac{ 1}{ \binom{ r}{ s}-1}$, then with high probability, $H^r(n,p)$ has a unique giant connected $s$-th-order component and its size is $(z+o(1)){ n\choose s}$, where $$z=1-\sum_{ j=0}^\infty \frac{ \left({ r\choose s}j -j+1 \right)^{ j-1}}{ j!}c^je^{ -c\left({ r\choose s}j -j+1\right)}.$$
Categories: math.CO