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題名 Asymptotic behavior for a generalized Domany Kinzel mode 作者 張書銓;陳隆奇;黃建豪
Chang, Shu-Chiuan;Chen, Lung-Chi;Huang, Chien-Hao貢獻者 應數系 關鍵詞 critical exponents and amplitudes; large deviation; percolation problems 日期 2017-02 上傳時間 12-Jul-2017 15:13:39 (UTC+8) 摘要 We consider a version of directed bond percolation on the square lattice 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, 2017, 023212+26 資料類型 article DOI http://dx.doi.org/10.1088/1742-5468/2017/2/023212 dc.contributor 應數系 dc.creator (作者) 張書銓;陳隆奇;黃建豪 zh-tw dc.creator (作者) Chang, Shu-Chiuan;Chen, Lung-Chi;Huang, Chien-Hao en-US dc.date (日期) 2017-02 dc.date.accessioned 12-Jul-2017 15:13:39 (UTC+8) - dc.date.available 12-Jul-2017 15:13:39 (UTC+8) - dc.date.issued (上傳時間) 12-Jul-2017 15:13:39 (UTC+8) - dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/111018 - dc.description.abstract (摘要) We consider a version of directed bond percolation on the square lattice 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 129 bytes - dc.format.mimetype text/html - dc.relation (關聯) Journal of Statistical Mechanics: Theory and Experiment, 2017, 023212+26 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 dc.doi.uri (DOI) http://dx.doi.org/10.1088/1742-5468/2017/2/023212
