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arXiv 2008.09143math.COmath.LO
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随机稀疏图和超图一阶性质的极限概率

Limiting probabilities of first order properties of random sparse graphs and hypergraphs

Alberto Larrauri, Tobias Müller, Marc Noy

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摘要：

令$G_n$是稀疏区域中的二项式随机图$G(n,p=c/n)$，众所周知，其在$c=1$处经历相变。Lynch（Random Structures Algorithms，1992）证明了对于每一个一阶句子$ϕ$，$G_n$满足$ϕ$为$n\to\infty$的极限概率存在，而且它是$c$的一个解析函数。本文考虑了$G_n$中一阶句子的所有极限概率的集合$L_c$在$[0,1]$中的闭包$\overline{L_c}$.我们发现，存在一个临界值$c_0 \approx0.93$，使$\overline{L_c}= [0,1]$时$c \ge c_0$，而$\overline{L_c}$错过至少一个子区间时，$c<c_0$。我们将这些结果推广到随机$d$ -均匀稀疏超图，其中超边的概率由$p=c/n^{d-1}$给出。

英文摘要：

Let $G_n$ be the binomial random graph $G(n,p=c/n)$ in the sparse regime, which as is well-known undergoes a phase transition at $c=1$. Lynch (Random Structures Algorithms, 1992) showed that for every first order sentence $ϕ$, the limiting probability that $G_n$ satisfies $ϕ$ as $n\to\infty$ exists, and moreover it is an analytic function of $c$. In this paper we consider the closure $\overline{L_c}$ in $[0,1]$ of the set $L_c$ of all limiting probabilities of first order sentences in $G_n$. We show that there exists a critical value $c_0 \approx0.93$ such that $\overline{L_c}= [0,1]$ when $c \ge c_0$, whereas $\overline{L_c}$ misses at least one subinterval when $c<c_0$. We extend these results to random $d$-uniform sparse hypergraphs, where the probability of a hyperedge is given by $p=c/n^{d-1}$.

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