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arXiv · 1103.2203

Asymptotic enumeration of independent sets on the Sierpinski gasket

Shu-Chiuan Chang·Lung-Chi Chen·Weigen Yan
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Abstract

The number of independent sets is equivalent to the partition function of the hard-core lattice gas model with nearest-neighbor exclusion and unit activity. We study the number of independent sets $m_{d,b}(n)$ on the generalized Sierpinski gasket $SG_{d,b}(n)$ at stage $n$ with dimension $d$ equal to two, three and four for $b=2$, and layer $b$ equal to three for $d=2$. The upper and lower bounds for the asymptotic growth constant, defined as $z_{SG_{d,b}}=\lim_{v \to \infty} \ln m_{d,b}(n)/v$ where $v$ is the number of vertices, on these Sierpinski gaskets are derived in terms of the results at a certain stage. The numerical values of these $z_{SG_{d,b}}$ are evaluated with more than a hundred significant figures accurate. We also conjecture the upper and lower bounds for the asymptotic growth constant $z_{SG_{d,2}}$ with general $d$.

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Shu-Chiuan Chang, Lung-Chi Chen, Weigen Yan. 2011-03-11. Asymptotic enumeration of independent sets on the Sierpinski gasket. https://arxiv.org/abs/1103.2203

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