TY - JOUR

T1 - Dimer coverings on the sierpinski gasket

AU - Chang, Shu Chiuan

AU - Chen, Lung Chi

N1 - Funding Information:
Acknowledgements We would like to thank Prof. D. Dhar for helpful discussions. The research of S.C.C. was partially supported by the NSC grant NSC-96-2112-M-006-001 and NSC-96-2119-M-002-001. The research of L.C.C was partially supported by TJ & MY Foundation and NSC grant NSC 96-2115-M-030-002.

PY - 2008/5

Y1 - 2008/5

N2 - We present the number of dimer coverings N d (n) on the Sierpinski gasket SG d (n) at stage n with dimension d equal to two, three, four or five. When the number of vertices, denoted as v(n), of the Sierpinski gasket is an even number, N d (n) is the number of close-packed dimers. When the number of vertices is an odd number, no close-packed configurations are possible and we allow one of the outmost vertices uncovered. The entropy of absorption of diatomic molecules per site, defined as S_SG_=n N_d(n)v(n) , is calculated to be ln∈(2)/3 exactly for SG 2. The numbers of dimers on the generalized Sierpinski gasket SG d,b (n) with d=2 and b=3,4,5 are also obtained exactly with entropies equal to ln∈(6)/7, ln∈(28)/12, ln∈(200)/18, respectively. The number of dimer coverings for SG 3 is given by an exact product expression, such that its entropy is given by an exact summation expression. The upper and lower bounds for the entropy are derived in terms of the results at a certain stage for SG 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_SG_{d}} with d=3,4,5 can be evaluated with more than a hundred significant figures accurate.

AB - We present the number of dimer coverings N d (n) on the Sierpinski gasket SG d (n) at stage n with dimension d equal to two, three, four or five. When the number of vertices, denoted as v(n), of the Sierpinski gasket is an even number, N d (n) is the number of close-packed dimers. When the number of vertices is an odd number, no close-packed configurations are possible and we allow one of the outmost vertices uncovered. The entropy of absorption of diatomic molecules per site, defined as S_SG_=n N_d(n)v(n) , is calculated to be ln∈(2)/3 exactly for SG 2. The numbers of dimers on the generalized Sierpinski gasket SG d,b (n) with d=2 and b=3,4,5 are also obtained exactly with entropies equal to ln∈(6)/7, ln∈(28)/12, ln∈(200)/18, respectively. The number of dimer coverings for SG 3 is given by an exact product expression, such that its entropy is given by an exact summation expression. The upper and lower bounds for the entropy are derived in terms of the results at a certain stage for SG 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_SG_{d}} with d=3,4,5 can be evaluated with more than a hundred significant figures accurate.

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U2 - 10.1007/s10955-008-9516-0

DO - 10.1007/s10955-008-9516-0

M3 - Article

AN - SCOPUS:41849094036

SN - 0022-4715

VL - 131

SP - 631

EP - 650

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 4

ER -