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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 5-Sep-2019 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
dc.contributor.advisor 陳隆奇zh_TW
dc.contributor.advisor Chen, Lung-Chien_US
dc.contributor.author (Authors) 林芳誼zh_TW
dc.contributor.author (Authors) Lin, Fang-Yien_US
dc.creator (作者) 林芳誼zh_TW
dc.creator (作者) Lin, Fang-Yien_US
dc.date (日期) 2019en_US
dc.date.accessioned 5-Sep-2019 16:13:36 (UTC+8)-
dc.date.available 5-Sep-2019 16:13:36 (UTC+8)-
dc.date.issued (上傳時間) 5-Sep-2019 16:13:36 (UTC+8)-
dc.identifier (Other Identifiers) G0106751001en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/125636-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用數學系zh_TW
dc.description (描述) 106751001zh_TW
dc.description.abstract (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。		zh_TW
dc.description.abstract (摘要) 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).		en_US
dc.description.tableofcontents 1 Introduction 1
2 Main results 5
3 Random walk 9
3.1 Derivation of D_n . . .. . . . . . . . . . . . . . . 9
3.2 Derivation of α_{n,c} .. . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Derivation of σ_n^2 . .. . . . . . . . . . . . . . . . . . . . . . . . 15
3.4 Behavior of α_{n,c} and σ_n as n → ∞ . . . . . . . . 17
3.4.1 The case that p_m ≈m→∞ p/m^s. .. . . . . . . . . . 17
3.4.2 The case that p_m = (e^(-λ)λ^m)/m! . . . . . . . . 21
4 The proof of main theorem 23
4.1 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . 23
4.2 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . 26
4.3 Proof of Theorem 2.4 . . . . . . . . . . . . . . . . 28
Bibliography 31		zh_TW
dc.format.extent 590351 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0106751001en_US
dc.subject (關鍵詞) Domany-Kinzel模型zh_TW
dc.subject (關鍵詞) 定向滲流zh_TW
dc.subject (關鍵詞) 隨機漫步zh_TW
dc.subject (關鍵詞) 漸進行為zh_TW
dc.subject (關鍵詞) 臨界值行為zh_TW
dc.subject (關鍵詞) Berry-Esseen定理zh_TW
dc.subject (關鍵詞) 大離差定理zh_TW
dc.subject (關鍵詞) Domany-Kinzel modelen_US
dc.subject (關鍵詞) Directed percolationen_US
dc.subject (關鍵詞) Random walken_US
dc.subject (關鍵詞) Asymptotic behavioren_US
dc.subject (關鍵詞) Critical behavioren_US
dc.subject (關鍵詞) Berry-Esseen theoremen_US
dc.subject (關鍵詞) Large deviationen_US
dc.title (題名) 向右之具長域Domany-Kinzel模型的漸進行為zh_TW
dc.title (題名) Asymptotic behavior for a long-range Domany-Kinzel model with right directionen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (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.		zh_TW
dc.identifier.doi (DOI) 10.6814/NCCU201900928en_US