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| 研究生： |
陳柏維 Chen, Bo-Wei |
|---|---|
| 論文名稱： |
二維謝爾賓斯基墊片上非有序性單體-二聚體的漸進行為 Asymptotic Behavior of Disordered Monomer-Dimer Model on Two Dimensional Sierpinski Gasket |
| 指導教授： |
陳隆奇
Chen, Lung-Chi |
| 口試委員: |
洪芷漪
Hong, Jyy-I 張書銓 Chang, Shu-Chuian |
| 學位類別： |
碩士
Master |
| 系所名稱： |
理學院 - 應用數學系 Department of Mathematical Sciences |
| 論文出版年： | 2025 |
| 畢業學年度： | 113 |
| 語文別： | 英文 |
| 論文頁數： | 28 |
| 中文關鍵詞： | 二聚體-單體模型 、謝爾賓斯基墊片 、遞迴關係 、漸近增長 |
| 外文關鍵詞： | Dimer-monomer model, Sierpinski gasket, Recursion relation, Asymptotic growth |
| 相關次數： | 點閱：49 下載：0 |
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我們考慮部署無序單體-二聚體在二維謝爾賓斯基墊片 $SG_n$ 上，並個別賦予單體及二聚體一個為正數的權重 $hi$ 和 $si$。我們研究該模型隨著 $n$ 增大時的漸近行為，並推導出配分函數的上界和下界。基於配分函數的上界和下界，估計配分函數的熵，推導隨著 $n$ 增大時熵的上下界且證明其收斂速度非常快。
In this thesis, we consider a disordered monomer-dimer model on the two-dimensional Sierpinski gasket at stage $n$ as $n\to\infty$, where we assign positive weights $\mathcal{m}$ and $\mathcal{d}$ to monomers and dimers, respectively. We investigate the asymptotic behavior of the model as $n$ grows and derive upper and lower bounds for the partition function with the rapid convergence rate. Furthermore, we provide estimations for the entropy of the partition function, which help to better understand the behavior of the model on the fractal structure.
中文摘要 i
Abstract ii
Contents iii
1 Introduction and Model 1
1.1 Introduction 1
1.2 Monomer-dimer model on two dimensional Sierpinski gasket 2
2 Main Result 5
2.1 Notations and the key lemmas 5
2.2 Main results 10
3 Proofs of the key lemmas 14
3.1 Proof of Lemma 2.2 14
3.2 Proof of Lemma 2.3 16
3.3 Proof of Lemma 2.4 19
4 Proof of the main results 21
4.1 Proof of Theorem 2.5 21
4.2 Proof of Theorem 2.6 23
4.3 Proof of Corollary 2.7 26
References 28
[1] Ole J Heilmann and Elliott H Lieb. Theory of monomer-dimer systems. Communications in mathematical Physics, 25(3):190–232, 1972.
[2] Weigen Yan and Yeong-Nan Yeh. On the matching polynomial of subdivision graphs. Discrete Applied Mathematics, 157(1):195–200, 2009.
[3] Alexandra Quitmann. Decay of correlations in the monomer-dimer model. Journal of Mathematical Physics, 65(10), 2024.
[4] F. Y. Wu, Wen-Jer Tzeng, and N. Sh. Izmailian. Exact solution of a monomer-dimer problem: A single boundary monomer on a nonbipartite lattice. Phys. Rev. E, 83:011106, Jan 2011.
[5] FY Wu. Erratum: Pfaffian solution of a dimer-monomer problem: Single monomer on the boundary [phys. rev. e 74, 020104 (r)(2006)]. Physical Review E— Statistical, Nonlinear, and Soft Matter Physics, 74(3):039907, 2006.
[6] Shu-Chiuan Chang and Lung-Chi Chen. Dimer-monomer model on the sierpinski gasket. Physica A: Statistical Mechanics and its Applications, 387(7):1551–1566, 2008.
[7] Partha S Dey and Kesav Krishnan. Disordered monomer-dimer model on cylinder graphs. Journal of Statistical Physics, 190(8):146, 2023.
[8] Shu-Chiuan Chang and Lung-Chi Chen. Asymptotic behavior of a generalized independent sets model on the two-dimensional sierpinski gasket. Journal of Mathematical Physics, 65(6), 2024.
全文公開日期 2026/06/24