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題名 Asymptotic behavior for a generalized Domany Kinzel mode 作者 張書銓
Chang, Shu-Chiuan
陳隆奇
Chen, Lung-Chi
黃建豪
Huang, Chien-Hao貢獻者 應數系 關鍵詞 critical exponents and amplitudes; large deviation;percolation problems 日期 2017 上傳時間 27-Jun-2017 17:09:11 (UTC+8) 摘要 We consider a version of directed bond percolation on the squarelattice such that horizontal edges are directed rightward with probabilities one, and vertical edges are directed upward with probabilities p 1, p 2 alternatively in even rows and probabilities p 2, p 1 alternatively in odd rows, where ${{p}_{1}}\\in \\left[0,1\\right)$ , ${{p}_{2}}\\in \\left[0,1\\right)$ , but ${{p}_{1}}\\vee {{p}_{2}}>0$ . Let $\\tau (M,N)$ be the probability that there is at least one connected-directed path of occupied edges from (0, 0) to (M,N). Defining the aspect ratio $\\alpha =M/N$ , we show that there is a critical value ${{\\alpha}_{c}}=\\left(2-{{p}_{1}}-{{p}_{2}}\\right)/\\left(\\,{{p}_{1}}+{{p}_{2}}\\right)$ such that as $N\\to \\infty $ , $\\tau (M,N)$ is 1, 0 and 1/2 for $\\alpha >{{\\alpha}_{c}}$ , $\\alpha <{{\\alpha}_{c}}$ and $\\alpha ={{\\alpha}_{c}}$ , respectively. In particular, the model reduces to the square lattice with uniform vertical probability when ${{p}_{1}}={{p}_{2}}$ [1], and the model reduces to the honeycomb lattice when one of p 1 and p 2 is equal to 0. We study how the critical value ${{\\alpha}_{c}}$ changes between the square lattice and the honeycomb lattice as bricks. In this article, we investigate the rate of convergence of $\\tau (M,N)$ and the asymptotic behavior of $\\tau \\left(M_{N}^{-},N\\right)$ and $\\tau \\left(M_{N}^{+},N\\right)$ , where $M_{N}^{-}/N\\uparrow {{\\alpha}_{c}}$ and $M_{N}^{+}/N\\downarrow {{\\alpha}_{c}}$ as $N\\uparrow \\infty $ . 關聯 Journal of Statistical Mechanics: Theory and Experiment, 資料類型 article dc.contributor 應數系 - dc.creator (作者) 張書銓 zh_TW dc.creator (作者) Chang, Shu-Chiuan - dc.creator (作者) 陳隆奇 zh_TW dc.creator (作者) Chen, Lung-Chi en_US dc.creator (作者) 黃建豪 zh_TW dc.creator (作者) Huang, Chien-Hao en_US dc.date (日期) 2017 - dc.date.accessioned 27-Jun-2017 17:09:11 (UTC+8) - dc.date.available 27-Jun-2017 17:09:11 (UTC+8) - dc.date.issued (上傳時間) 27-Jun-2017 17:09:11 (UTC+8) - dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/110507 - dc.description.abstract (摘要) We consider a version of directed bond percolation on the squarelattice such that horizontal edges are directed rightward with probabilities one, and vertical edges are directed upward with probabilities p 1, p 2 alternatively in even rows and probabilities p 2, p 1 alternatively in odd rows, where ${{p}_{1}}\\in \\left[0,1\\right)$ , ${{p}_{2}}\\in \\left[0,1\\right)$ , but ${{p}_{1}}\\vee {{p}_{2}}>0$ . Let $\\tau (M,N)$ be the probability that there is at least one connected-directed path of occupied edges from (0, 0) to (M,N). Defining the aspect ratio $\\alpha =M/N$ , we show that there is a critical value ${{\\alpha}_{c}}=\\left(2-{{p}_{1}}-{{p}_{2}}\\right)/\\left(\\,{{p}_{1}}+{{p}_{2}}\\right)$ such that as $N\\to \\infty $ , $\\tau (M,N)$ is 1, 0 and 1/2 for $\\alpha >{{\\alpha}_{c}}$ , $\\alpha <{{\\alpha}_{c}}$ and $\\alpha ={{\\alpha}_{c}}$ , respectively. In particular, the model reduces to the square lattice with uniform vertical probability when ${{p}_{1}}={{p}_{2}}$ [1], and the model reduces to the honeycomb lattice when one of p 1 and p 2 is equal to 0. We study how the critical value ${{\\alpha}_{c}}$ changes between the square lattice and the honeycomb lattice as bricks. In this article, we investigate the rate of convergence of $\\tau (M,N)$ and the asymptotic behavior of $\\tau \\left(M_{N}^{-},N\\right)$ and $\\tau \\left(M_{N}^{+},N\\right)$ , where $M_{N}^{-}/N\\uparrow {{\\alpha}_{c}}$ and $M_{N}^{+}/N\\downarrow {{\\alpha}_{c}}$ as $N\\uparrow \\infty $ . - dc.format.extent 1444544 bytes - dc.format.mimetype application/pdf - dc.relation (關聯) Journal of Statistical Mechanics: Theory and Experiment, - dc.subject (關鍵詞) critical exponents and amplitudes; large deviation;percolation problems - dc.title (題名) Asymptotic behavior for a generalized Domany Kinzel mode - dc.type (資料類型) article - dc.identifier.doi (DOI) 10.1088/1742-5468/2017/2/023212 -
