Dimer-Monomer Model on the Sierpinski Gasket

Abstract

We present the numbers of dimer–monomers Md(n)Md(n) on the Sierpinski gasket SGd(n)SGd(n) at stage nn with dimension dd equal to two, three and four. The upper and lower bounds for the asymptotic growth constant, defined as zSGd=limv→∞lnMd(n)/vzSGd=limv→∞lnMd(n)/v where vv is the number of vertices on SGd(n)SGd(n), are derived in terms of the results at a certain stage. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of zSGdzSGd can be evaluated with more than a hundred significant figures accurate. From the results for d=2,3,4d=2,3,4, we conjecture the upper and lower bounds of zSGdzSGd for general dimension. The corresponding results on the generalized Sierpinski gasket SGd,b(n)SGd,b(n) with d=2d=2 and b=3,4b=3,4 are also obtained.