Dimer coverings on the Sierpinski gasket with possible vacancies on the outmost vertices

Abstract

We present the number of dimers $N_d(n)$ on the Sierpinski gasket $S{G}_d(n)$ at stage $n$ with dimension $d$ equal to two, three, four or five, where one of the outmost vertices is not covered when the number of vertices $v(n)$ is an odd number. The entropy of absorption of diatomic molecules per site, defined as $S_{S{G}_d}={\lim }_{n\to \infty }\ln N_d(n)/v(n)$, is calculated to be $\ln (2)/3$ exactly for $S{G}_2(n)$. The numbers of dimers on the generalized Sierpinski gasket $S{G}_{d,b}(n)$ with $d=2$ and $b=3,4,5$ are also obtained exactly. Their entropies are equal to $\ln (6)/7$, $\ln (28)/12$, $\ln (200)/18$, respectively. The upper and lower bounds for the entropy are derived in terms of the results at a certain stage for $S{G}_d(n)$ with $d=3,4,5$. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of $S_{S{G}_d}$ with $d=3,4,5$ can be evaluated with more than a hundred significant figures accurate.

35 pages, 20 figures and 1 table

References in corpus (4)

• Dimer coverings on the Sierpinski gasket with possible vacancies on the outmost vertices
• Dimer-monomer model on the Sierpinski gasket
• The Pfaffian solution of a dimer-monomer problem: Single monomer on the boundary
• Branched Polymers on the Given-Mandelbrot family of fractals

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• Ice model and eight-vertex model on the two-dimensional Sierpinski gasket
• Average Trapping time on the 3-dimensional 3-level Sierpinski gasket network with a set of trap nodes
• Potts Partition Function Zeros and Ground State Entropy on Hanoi Graphs
• Perfect matchings of line graphs with small maximum degree