Asymptotic behavior of a generalized independent sets model on the two-dimensional Sierpinski gasket

Abstract

We consider a generalized independent set model on the two-dimensional Sierpinski gasket SGn. Let A be a set of symbols and each vertex of SGn should take a symbol from A. Given a subset S⊂A such that the symbols of nearest-neighbor vertices are not allowed to be the forbidden blocks F={ij:i,j∈S}. The probability of each vertex of SGn to take a symbol in S is denoted by p ∈ (0, 1), and the probability of each vertex to take a symbol not in S is given by 1 − p. When A={0,1}, S={0}, and p = 1/2, this reduces to independent set model or golden mean shift in symbolic dynamics. We investigate the asymptotic behavior of this generalized model on the two-dimensional Sierpinski gasket and obtain upper and lower bounds of the entropy per site for p ∈ (0, 1).