謝爾賓斯基墊片上一般獨立集數量的漸近行為

Asymptotic enumeration of general independent sets on the Sierpinski gasket

陳品樺(Chen Pin-Hua)

指導教授 ： 陳隆奇

國立政治大學/理學院/應用數學系/碩士(2022年)

https://doi.org/10.6814/NCCU202201726

摘要

計算獨立集的數量一直都是個重要且有趣的問題，在二維整數晶格中，
獨立集的數量上熵的極限存在，並估計其上下界，已被 [13] 探討。此外在
二維與三維謝爾賓斯基墊片獨立集的數量上熵的極限存在，並估計其上下
界也被 [32] 探討。
在本篇論文中，我們將推廣 [32] 二維的謝爾賓斯基墊片的獨立集個數
到機率分布，其解釋如下：令 A 是一個有限符號集合，並給定任意子集
S ⊂ A，假設 p ∈ (0, 1) 是二維的謝爾賓斯基墊片裡的每一個點屬於 S 的機率，又令 F = {ij : ij ∈ E, i, j ∈ S}，其中 E 是謝爾賓斯基墊片的邊，這是一種不被允許出現的情況。舉個特例，當 A = {0, 1}, S = {0} 而且 p = 1/2 (均勻分布) 在這種情況之下，就是 [32] 這篇文章的模型。這篇文章我們將研究這個分布模型 (p ∈ (0, 1)) 中熵的漸近行為。

關鍵字

謝爾賓斯基墊片 ； 獨立集

並列摘要

Enumerate the number of independent sets or golden mean shift on the graph is a very interesting and important problem. For the square lattice, it was shown that the limit of the entropy of the number of independent sets exists and its upper and lower bounds were estimated [13], and its had been considered on various graphs [15]–[17]. Moreover, it is of interest to consider independent sets on self-similar fractal lattices which have scaling invariance rather than translational invariance [18]. Sierpinski gasket is a kind of self-similar fractal lattices, and the number of independent sets on Sierpinski gasket had been investigated in [32]. In
this thesis, we extend the independent sets in [32] to be more general model on the two-dimensional Sierpinski gasket. Let A is a set of symbol and given any S ⊂ A. Let p ∈ (0, 1) be the probability of each vertex of SG(n) with a symbol
that is in symbols S, and 1 − p be the probability of each vertex of SG(n) with a symbol that is not in symbols S where F = {ij : i, j ∈ S} is a forbidden blocks between nearest-neighbor vertices. In particular, it is so called a golden mean shift or independent set when A = {0, 1}, S = {0} and p = 1/2 . We will investigate the asymptotes behavior of the entropy in this general model.

並列關鍵字

Sierpinski gasket ； Independent sets

參考文獻

[1] L. K. Runnels, Phase transitions of hard sphere lattice gases, Communications in Mathematical Physics 40 (1975) 37–48.

[2] G. R. Brightwell, O. Häggström, P. Winkler, Nonmonotonic behavior in hard-core and Widom-Rowlinson models, Journal of Statistical Physics 94 (1999) 415–435.

[3] J. R. Heringa, H. W. J. Blöte, E. Luijten, High-dimensional lattice gases, Journal of Physics

A-Mathematical and General 33 (2000) 2929–2941.

[4] W. Guo, H. W. Blöte, Finite-size analysis of the hard-square lattice gas, Physical Review E 66 (2002) 046140.