离散超立方体中随机独立集的阈值现象
A threshold phenomenon for random independent sets in the discrete hypercube
摘要：
令$I$是根据具有参数$λ>0$的硬核分布（即，其中每个独立集合$I$以与$λ^{|I|}$成比例的概率被选择的分布）从离散$d$维超立方体$Q_d=\{0,1\}^d$中提取的独立集合。我们发现$I$的出现在$λ=1$周围有一个急剧的转变：对于$λ>1$，$\min\{|I \cap {\cal E}|, |I \cap {\cal O}|\}=0$渐近几乎必然，其中${\cal E}$和${\cal O}$是$Q_d$的二分类，而对于$λ<1$，$\min\{|I \cap {\cal E}|, |I \cap {\cal O}|\}$在$d$中渐近几乎必然是指数的。跃迁发生在一个长度为$1/d$阶的区间内。证明中的一个关键步骤是估计$Z_λ(Q_d)$，$Q_d$中的独立集合的总和，其中每个集合$I$给定权重$λ^{|I|}$（a.k.a.核心分区函数）。我们得到了$Z_λ(Q_d)$对$λ>\sqrt{2}-1$的渐近性，以及对$λ\leq \sqrt{2}-1$的近似匹配的上、下界，推广了Korshunov和Sapozhenko的工作.这些界限允许我们读出一些关于根据硬核分布绘制的独立集合的结构的非常具体的信息。我们还得到了一个长期影响的结果。对于所有固定的$λ>0$，如果$I$是根据具有参数$λ$的硬核分布从$Q_d$的独立集合中选择的，条件是特定的$v \in {\cal E}$在$I$中，则另一个顶点$w$在$I$中的概率对于$w \in {\cal O}$是$o(1)$，但是对于$w \in {\cal E}$是$Ω(1)$。
英文摘要：
Let $I$ be an independent set drawn from the discrete $d$-dimensional hypercube $Q_d=\{0,1\}^d$ according to the hard-core distribution with parameter $λ>0$ (that is, the distribution in which each independent set $I$ is chosen with probability proportional to $λ^{|I|}$). We show a sharp transition around $λ=1$ in the appearance of $I$: for $λ>1$, $\min\{|I \cap {\cal E}|, |I \cap {\cal O}|\}=0$ asymptotically almost surely, where ${\cal E}$ and ${\cal O}$ are the bipartition classes of $Q_d$, whereas for $λ<1$, $\min\{|I \cap {\cal E}|, |I \cap {\cal O}|\}$ is asymptotically almost surely exponential in $d$. The transition occurs in an interval whose length is of order $1/d$. A key step in the proof is an estimation of $Z_λ(Q_d)$, the sum over independent sets in $Q_d$ with each set $I$ given weight $λ^{|I|}$ (a.k.a. the hard-core partition function). We obtain the asymptotics of $Z_λ(Q_d)$ for $λ>\sqrt{2}-1$, and nearly matching upper and lower bounds for $λ\leq \sqrt{2}-1$, extending work of Korshunov and Sapozhenko. These bounds allow us to read off some very specific information about the structure of an independent set drawn according to the hard-core distribution. We also derive a long-range influence result. For all fixed $λ>0$, if $I$ is chosen from the independent sets of $Q_d$ according to the hard-core distribution with parameter $λ$, conditioned on a particular $v \in {\cal E}$ being in $I$, then the probability that another vertex $w$ is in $I$ is $o(1)$ for $w \in {\cal O}$ but $Ω(1)$ for $w \in {\cal E}$.
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