Please use this identifier to cite or link to this item:
https://ah.lib.nccu.edu.tw/handle/140.119/125636
DC Field | Value | Language |
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dc.contributor.advisor | 陳隆奇 | zh_TW |
dc.contributor.advisor | Chen, Lung-Chi | en_US |
dc.contributor.author | 林芳誼 | zh_TW |
dc.contributor.author | Lin, Fang-Yi | en_US |
dc.creator | 林芳誼 | zh_TW |
dc.creator | Lin, Fang-Yi | en_US |
dc.date | 2019 | en_US |
dc.date.accessioned | 2019-09-05T08:13:36Z | - |
dc.date.available | 2019-09-05T08:13:36Z | - |
dc.date.issued | 2019-09-05T08:13:36Z | - |
dc.identifier | G0106751001 | en_US |
dc.identifier.uri | http://nccur.lib.nccu.edu.tw/handle/140.119/125636 | - |
dc.description | 碩士 | zh_TW |
dc.description | 國立政治大學 | zh_TW |
dc.description | 應用數學系 | zh_TW |
dc.description | 106751001 | zh_TW |
dc.description.abstract | 在本篇文章中,我們介紹一種向右之具長域的Domany-Kinzel 模型,其\n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。 | zh_TW |
dc.description.abstract | 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\nis 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). | en_US |
dc.description.tableofcontents | 1 Introduction 1\n2 Main results 5\n3 Random walk 9\n3.1 Derivation of D_n . . .. . . . . . . . . . . . . . . 9\n3.2 Derivation of α_{n,c} .. . . . . . . . . . . . . . . . . . . . . . . . . 11\n3.3 Derivation of σ_n^2 . .. . . . . . . . . . . . . . . . . . . . . . . . 15\n3.4 Behavior of α_{n,c} and σ_n as n → ∞ . . . . . . . . 17\n3.4.1 The case that p_m ≈m→∞ p/m^s. .. . . . . . . . . . 17\n3.4.2 The case that p_m = (e^(-λ)λ^m)/m! . . . . . . . . 21\n4 The proof of main theorem 23\n4.1 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . 23\n4.2 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . 26\n4.3 Proof of Theorem 2.4 . . . . . . . . . . . . . . . . 28\nBibliography 31 | zh_TW |
dc.format.extent | 590351 bytes | - |
dc.format.mimetype | application/pdf | - |
dc.source.uri | http://thesis.lib.nccu.edu.tw/record/#G0106751001 | en_US |
dc.subject | Domany-Kinzel模型 | zh_TW |
dc.subject | 定向滲流 | zh_TW |
dc.subject | 隨機漫步 | zh_TW |
dc.subject | 漸進行為 | zh_TW |
dc.subject | 臨界值行為 | zh_TW |
dc.subject | Berry-Esseen定理 | zh_TW |
dc.subject | 大離差定理 | zh_TW |
dc.subject | Domany-Kinzel model | en_US |
dc.subject | Directed percolation | en_US |
dc.subject | Random walk | en_US |
dc.subject | Asymptotic behavior | en_US |
dc.subject | Critical behavior | en_US |
dc.subject | Berry-Esseen theorem | en_US |
dc.subject | Large deviation | en_US |
dc.title | 向右之具長域Domany-Kinzel模型的漸進行為 | zh_TW |
dc.title | Asymptotic behavior for a long-range Domany-Kinzel model with right direction | en_US |
dc.type | thesis | en_US |
dc.relation.reference | [1] B. Bollobas and O. Riordan. Percolation. Cambridge University Press, 2006.\n[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.\n[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.\n[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.\n[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.\n[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\nExperiment, 2017(2):023212, feb 2017.\n[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.\n[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.\n[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.\n[10] Ben T Graham. Sublinear variance for directed last-passage percolation. Journal of Theoretical Probability, 25(3):687–702, 2012.\n[11] Geoffrey R. Grimmett. Percolation, volume 321 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag Berlin Heidelberg, 2 edition, 1999.\n[12] Malte Henkel, Haye Hinrichsen, and Sven Lübeck. Non-Equilibrium Phase Transitions. Springer Netherlands, 1 edition, 2008.\n[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.\n[14] Hugo Touchette. Physics Reports, volume 478. 2009.\n[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–\n778, Mar 1982. | zh_TW |
dc.identifier.doi | 10.6814/NCCU201900928 | en_US |
item.cerifentitytype | Publications | - |
item.openairecristype | http://purl.org/coar/resource_type/c_46ec | - |
item.openairetype | thesis | - |
item.fulltext | With Fulltext | - |
item.grantfulltext | restricted | - |
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