Asymptotic Enumeration of Independent Sets on the Sierpinski Gasket

Shu-Chiuan Chang, Lung-Chi Chen, Weigen Yan

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 $d=2$. Upper and lower bounds for the asymptotic growth constant, defined as $Z_{SG_{d,b}}=lim_{v\rightarrow \infty ln m_{d,b} (n)/v$ where $v$ is the number of vertices, on these Sierpinski gaskets are derived in terms of the numbers 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 upper and lower bounds for the asymptotic growth constant $Z_{SG_{d,2}}$ with general $d$, and an approximation of $Z_{SG_{d,2}}$ when $d$ is large.