Growth exponent in the Domany-Kinzel cellular automaton

In a roughening process, the growth exponent ß describes how the roughness w grows with the time t: w ~ tß. We determine the exponent ß of a growth process generated by the spatiotemporal patterns of the one-dimensional Domany-Kinzel cellular automaton. The values obtained for ß show a cusp at the frozen/active transition which permits determination of the transition line. The ß value at the transition depends on the scheme used: symmetric \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\beta \simeq 0.83)$$ \end{document} or non-symmetric \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\beta \simeq 0.61)$$ \end{document}. Using damage spreading ideas, we also determine the active/chaotic transition line; this line depends on how the replicas are updated.