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| Title: | 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 |
| Authors: | 林芳誼
Lin, Fang-Yi |
| Contributors: | 陳隆奇
Chen, Lung-Chi
林芳誼
Lin, Fang-Yi |
| Keywords: | 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 |
| Date: | 2019 |
| Issue Date: | 2019-09-05 16:13:36 (UTC+8) |
| 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
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). |
| Reference: | [1] B. Bollobas and O. Riordan. Percolation. Cambridge University Press, 2006.
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[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.
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[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. |
| Description: | 碩士
國立政治大學
應用數學系
106751001 |
| Source URI: | http://thesis.lib.nccu.edu.tw/record/#G0106751001 |
| Data Type: | thesis |
| DOI: | 10.6814/NCCU201900928 |
| Appears in Collections: | [應用數學系] 學位論文
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