论文标题
在HyperCube的随机子图中独立集
Independent sets in random subgraphs of the hypercube
论文作者
论文摘要
令$ q_ {d，p} $为$ d $ - 二维超模型$ \ {0,1 \}^d $的随机子图，其中每个边缘都以概率$ p $独立保留。我们研究了$ q_ {d，p} $中的独立集的渐近数，为$ d \ to \ infty $，用于广泛的参数$ p $，包括$ p $的值，$ p $的值趋于零的速度为$ \ frac {c \ log d} {c \ log d} {d^{d^{1/3}} $，不在$ p $ $ p $ $ p $ p $ p $ p $ p $ p $ p $结果扩展到$ q_ {d，p} $的硬核模型，并通过研究超数据管上的密切相关的反铁磁ising模型获得，可以将其视为超平方上的正温核心模型。这些结果概括了Galvin，Jenssen和Perkins在HyperCube上的硬核模型上的先前结果，与$ P = 1 $相对应，这扩展了Korshunov和Sapozhenko在HyperCube中渐近数的独立集中的经典结果。
Let $Q_{d,p}$ be the random subgraph of the $d$-dimensional hypercube $\{0,1\}^d$, where each edge is retained independently with probability $p$. We study the asymptotic number of independent sets in $Q_{d,p}$ as $d \to \infty$ for a wide range of parameters $p$, including values of $p$ tending to zero as fast as $\frac{C\log d}{d^{1/3}}$, constant values of $p$, and values of $p$ tending to one. The results extend to the hardcore model on $Q_{d,p}$, and are obtained by studying the closely related antiferromagnetic Ising model on the hypercube, which can be viewed as a positive-temperature hardcore model on the hypercube. These results generalize previous results by Galvin, Jenssen and Perkins on the hard-core model on the hypercube, corresponding to the case $p=1$, which extended Korshunov and Sapozhenko's classical result on the asymptotic number of independent sets in the hypercube.
