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| 研究生： |
康瑋翔 Kang, Wei-Hsiang |
|---|---|
| 論文名稱： |
謝爾賓斯基墊片上邊滲流中特定無限簇的漸近行為 Asymptotic Behavior of a Version of Infinite Cluster of the Bond Percolation Model on the 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-Chiuan |
| 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： | 2026 |
| 畢業學年度： | 115 |
| 語文別： | 英文 |
| 論文頁數： | 27 |
| 中文關鍵詞： | 謝爾賓斯基墊片 、限滲透模型 、精確熵 |
| 外文關鍵詞： | Sierpinski gasket, Restricted percolation model, Exact entropy |
| 相關次數： | 點閱：66 下載：3 |
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在本論文中，我們探討定義於謝爾賓斯基墊片上的受限滲透模型。有別於標準的滲透理論，我們專注於研究一個特殊的無窮連通叢集，其嚴格限制該叢集內的每一個頂點至多只能擁有一條封閉邊。針對此模型，我們分析系統在每一階段 n 的遞迴與漸近行為。本研究的核心重點在於該系統的精確熵：透過在每一階段推導其嚴格的上下界，我們據此確立並推論了模型在無窮極限下的行為。最後，我們進一步檢視並量化了這些參數的收斂速率。
In this thesis, we investigate a restricted percolation model defined on the Sierpinski gasket. Diverging from standard percolation theory, we focus on a special infinite connected cluster subject to the strict constraint that each vertex within the cluster is incident to at most one closed edge. For this model, we analyze the recursive and asymptotic behaviors of the system at each stage n. The central focus of this research is the exact entropy of the system: by deriving rigorous upper and lower bounds at each stage, we establish and deduce the behavior of the model in the infinite limit. Finally, we further examine and quantify the convergence rates of these parameters.
中文摘要 i
Abstract ii
Contents iii
1 Introduction and Preliminaries 1
1.1 Introduction 1
1.2 Preliminaries 4
2 Main Result 6
2.1 Notations and the key lemmas 6
2.2 Main results 9
3 Proof of the main results 13
3.1 Proof of Theorem 2.4 13
3.2 Proof of Theorem 2.5 15
3.3 Proof of Theorem 2.6 18
3.4 Proof of Corollary 2.7 21
4 Proofs of the key lemmas 23
4.1 Proof of Lemma 2.2 23
4.2 Proof of Lemma 2.3 25
References 26
[1] M. V. Menshikov. Coincidence of critical points in percolation problems. Soviet Mathematics, Doklady, 33:856–859, 1986.
[2] M. Aizenman and D. J. Barsky. Sharpness of the phase transition in ercolation models.Communications in Mathematical Physics, 108:489–526, 1987.
[3] H. Duminil-Copin and V. Tassion. A new proof of the sharpness of the phase transition for bernoulli percolation and the ising model. Communications in Mathematical Physics,343:725–745, 2016.
[4] G. Grimmett. Percolation. Springer, 2nd edition, 1999.
[5] T. Hara and G. Slade. Self-avoiding walk in five or more dimensions. i. the critical behaviour. Communications in Mathematical Physics, 147:101–136, 1992.
[6] N. J. Calkin and H. S. Wilf. The number of independent sets in a grid graph. SIAM Journal on Discrete Mathematics, 11:54–60, 1998.
[7] R. J. Baxter. Planar lattice gases with nearest-neighbor exclusion. Annals of Combinatorics, 3:191–203, 1999.
[8] V. Linek. Bipartite graphs can have any number of independent sets. Discrete Mathematics,76:131–136, 1989.
[9] Y. Zhao. The number of independent sets in a regular graph. Combinatorics, Probability and Computing, 19:315–320, 2010.
[10] E. Teufl and S. Wagner. Enumeration problems for classes of self-similar graphs. Journal of Combinatorial Theory Series A, 114:1254–1277, 2007.
[11] B. B. Mandelbrot. The Fractal Geometry of Nature. Freeman, San Francisco, 1982.
[12] K. J. Falconer. Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester, 2nd edition, 2003.
[13] Y. Gefen, B. B. Mandelbrot, and A. Aharony. Critical phenomena on fractal lattices. Physical Review Letters, 45:855–858, 1980.
[14] S.-C. Chang and L.-C. Chen. Hamiltonian walks on the sierpinski gasket. Journal of Mathematical Physics, 52:023301, 2011.
[15] S.-C. Chang and L.-C. Chen. Asymptotic enumeration of independent sets on the sierpinski gasket. Filomat, 27(1):23–40, 2013.
[16] S.-C. Chang and L.-C. Chen. Asymptotic behavior of a generalized independent sets model on the two-dimensional sierpinski gasket. Journal of Mathematical Physics, 65:063301,2024.
[17] Masato Shinoda. Percolation on the pre-sierpinski gasket. Osaka Journal of Mathematics, 33(2):533–554, 1996
