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| Title: | Asymptotic behavior for a generalized Domany Kinzel mode |
| Authors: | 張書銓;陳隆奇;黃建豪
Chang, Shu-Chiuan;Chen, Lung-Chi;Huang, Chien-Hao |
| Contributors: | 應數系 |
| Keywords: | critical exponents and amplitudes;large deviation;percolation problems |
| Date: | 2017-02 |
| Issue Date: | 2017-07-12 15:13:39 (UTC+8) |
| Abstract: | 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 $ . |
| Relation: | Journal of Statistical Mechanics: Theory and Experiment, 2017, 023212+26 |
| Data Type: | article |
| DOI 連結: | http://dx.doi.org/10.1088/1742-5468/2017/2/023212 |
| DOI: | 10.1088/1742-5468/2017/2/023212 |
| Appears in Collections: | [應用數學系] 期刊論文
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