政大機構典藏-National Chengchi University Institutional Repository(NCCUR):Item 140.119/125636

政大機構典藏-National Chengchi University Institutional Repository(NCCUR):Item 140.119/125636

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政大機構典藏 > 理學院 > 應用數學系 > 學位論文 > Item 140.119/125636

請使用永久網址來引用或連結此文件: https://nccur.lib.nccu.edu.tw/handle/140.119/125636

題名:	向右之具長域Domany-Kinzel模型的漸進行為 Asymptotic behavior for a long-range Domany-Kinzel model with right direction
作者:	林芳誼 Lin, Fang-Yi
貢獻者:	陳隆奇 Chen, Lung-Chi 林芳誼 Lin, Fang-Yi
關鍵詞:	Domany-Kinzel模型定向滲流隨機漫步漸進行為臨界值行為 Berry-Esseen定理大離差定理 Domany-Kinzel model Directed percolation Random walk Asymptotic behavior Critical behavior Berry-Esseen theorem Large deviation
日期:	2019
上傳時間:	2019-09-05 16:13:36 (UTC+8)
摘要:	在本篇文章中，我們介紹一種向右之具長域的Domany-Kinzel 模型，其模型定義在二維方格座標上，假設n為一個非負整數，每個座標點(a, b) 都擁有具機率一的向右有向鏈結，並擁有n + 1 個分別具有p_k ∈ (0, 1)機率的從(a, b)到(a+k, b+1)之有向鏈結，其中a, b ∈ Z+ 且k = 0, 1, · · · , n。假設τ_n(N,M) 為從(0, 0) 到(N,M) 至少有一個由被滲透的邊組成之連通的有向路徑之機率，定義長寬比以α = N/M 表示，我們求得臨界值α_{n,c} ∈ R+ 使得當α = α_{n,c} 時在M趨近於無限下τ_n(N,M)趨近於1/2，並對其收斂速率進行研討。進而我們研究對n 趨近於無限時模型的表現，在m 為非負整數且p_m ∈ [0, 1) 的前提下，特別聚焦於p_m ≈m→∞ p/m^s其中p ∈ (0, 1)、s > 1，以及p_m=(e^(-λ)λ^m)/m!，這兩種假設情況進行討論，我們發現當s和λ的值符合前述情境時，lim_{n→∞} τ_n(N,M) 的極值表現與先前n為非負整數時的結果相似，並且在n趨近於無限的模型中，lim_{n→∞} τ_n(N,M) 的極值表現受α逼近α_{n,c} 的速度影響甚劇。 In this thesis, we introduce a certain type of Domany-Kinzel model which may be regarded as a long-range model with right direction in two-dimension rectangular lattices. For a fixed non-negative integer n, every site (a, b) possesses not only a directed bond from site (a, b) to (a + 1, b) with probability one but also n + 1 directed bonds from (a, b) to (a + k, b + 1) with respectively probabilities p_k ∈ (0, 1), ∀a, b ∈ Z+, k = 0, 1 · · · n. Let τ_n(N,M) be the probability that there is at least one connected-directed path of occupied edges from (0, 0) to (N,M) and let α be the aspect ratio which means α = N/M. We conclude that τ_n(N,M) converges to 1, 0, and 1/2 as M → ∞ for α > α_{n,c}, α < α_{n,c}, and α = α_{n,c}, respectively, where α_{n,c} ∈ R+ is the critical value. The rate of convergence is discussed, too. Moreover, we study the cases that n tends to infinity. Specifically, for p_m ∈ [0, 1) with m ∈ Z+, we discuss the two cases in detail which are p_m ≈m→∞ p/m^s with p ∈ (0, 1), s > 1 and p_m=(e^(-λ)λ^m)/m! with λ > 0. We discover that the behavior of lim_{n→∞} τ_n(N,M) is similar to the case that n is a non-negative integer when s and λ fit the definition. Moreover, the speed of α approaching to the critical apect ratio highly influences the behavior of lim_{n→∞} τ_n(N,M).
參考文獻:	[1] B. Bollobas and O. Riordan. Percolation. Cambridge University Press, 2006. [2] Simon R Broadbent and John M Hammersley. Percolation processes: I. crystals and mazes. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 53, pages629–641. Cambridge University Press, 1957. [3] Shu-Chiuan Chang and Lung-Chi Chen. Asymptotic behavior for a version of directed percolation on the triangular lattice. Journal of Statistical Physics, 155(3):500–522, May 2014. [4] Shu-Chiuan Chang and Lung-Chi Chen. Asymptotic behavior for a version of directed percolation on the honeycomb lattice. Physica A: Statistical Mechanics and its Applications, 436:547 – 557, 2015. [5] Shu-Chiuan Chang and Lung-Chi Chen. Asymptotic behavior for a long-range domany–kinzel model. Physica A: Statistical Mechanics and its Applications, 506:112 – 127, 2018. [6] Shu-Chiuan Chang, Lung-Chi Chen, and Chien-Hao Huang. Asymptotic behavior for a generalized domany–kinzel model. Journal of Statistical Mechanics: Theory and Experiment, 2017(2):023212, feb 2017. [7] Lung-Chi Chen. Asymptotic behavior for a version of directed percolation on a square lattice. Physica A: Statistical Mechanics and its Applications, 390(3):419 – 426, 2011. [8] Eytan Domany and Wolfgang Kinzel. Directed percolation in two dimensions: Numerical analysis and an exact solution. Phys. Rev. Lett., 47:5–8, Jul 1981. [9] Carl-Gustav Esseen. On the liapunoff limit of error in the theory of probability. Arkiv för Matematik, Astronomi och Fysik, A 28:1–19, 1942. [10] Ben T Graham. Sublinear variance for directed last-passage percolation. Journal of Theoretical Probability, 25(3):687–702, 2012. [11] Geoffrey R. Grimmett. Percolation, volume 321 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag Berlin Heidelberg, 2 edition, 1999. [12] Malte Henkel, Haye Hinrichsen, and Sven Lübeck. Non-Equilibrium Phase Transitions. Springer Netherlands, 1 edition, 2008. [13] T. C. Li and Z. Q. Zhang. A long-range domany-kinzel model of directed percolation. Journal of Physics A: Mathematical and General, 16(12):L401–L406, Aug 1983. [14] Hugo Touchette. Physics Reports, volume 478. 2009. [15] F. Y. Wu and H. Eugene Stanley. Domany-kinzel model of directed percolation: Formulation as a random-walk problem and some exact results. Phys. Rev. Lett., 48:775– 778, Mar 1982.
描述:	碩士國立政治大學應用數學系 106751001
資料來源:	http://thesis.lib.nccu.edu.tw/record/#G0106751001
資料類型:	thesis
DOI:	10.6814/NCCU201900928
顯示於類別:	[應用數學系] 學位論文

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