论文标题
二项式随机超图的渐近线性通过簇依赖性下的簇扩展
Asymptotic linearity of binomial random hypergraphs via cluster expansion under graph-dependence
论文作者
论文摘要
让整数$ n \ ge 3 $和整数$ r = r（n）\ ge 3 $。将二项式随机$ r $ r $ r $ r $ rusthypraph $ h_r（n，p）$定义为顶点套装$ [n] $上的$ r $ - 均匀图，使每个$ r $ $ set都是一个独立的边缘，具有概率$ p $。如果每对Hyperedges在最多一个顶点相交，则超图是线性的。我们通过群集扩展研究了随机超图$ h_r（n，p）$的线性概率，并提供有关所讨论概率的更精确的渐近渐进性，从而提高了麦凯和蒂恩获得的线性渐近概率，尤其是$ r = 3 $和$ r = 3 $和$ p = o（n^和$ p = o（n^o（n^n^{-7/7/5}））$。
Let integer $n \ge 3$ and integer $r = r(n) \ge 3$. Define the binomial random $r$-uniform hypergraph $H_r(n, p)$ to be the $r$-uniform graph on the vertex set $[n]$ such that each $r$-set is an edge independently with probability $p$. A hypergraph is linear if every pair of hyperedges intersects in at most one vertex. We study the probability of linearity of random hypergraphs $H_r(n, p)$ via cluster expansion and give more precise asymptotics of the probability in question, improving the asymptotic probability of linearity obtained by McKay and Tian, in particular, when $r=3$ and $p = o(n^{-7/5})$.
