向右之具長域Domany-Kinzel模型的漸進行為

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。

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).