Asymptotic enumeration of independent sets on the Sierpinski gasket

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 md,b(n) on the generalized Sierpinski gasket SGd,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 ZSGdb = limu(∞ ln md,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 ZSGdb are evaluated with more than a hundred significant figures accurate. We also conjecture upper and lower bounds for the asymptotic growth constant ZSGd2 with general d, and an approximation of ZSGd2 when d is large.