Growth Exponent in the Domany-Kinzel Cellular Automaton

In a roughening process, the growth exponent $β$ describes how the roughness $w$ grows with the time $t$: $w\sim t^β$. We determine the exponent $β$ of a growth process generated by the spatiotemporal patterns of the one dimensional Domany-Kinzel cellular automaton. The values obtained for $β$ shows a cusp at the frozen/active transition which permits determination of the transition line. The $β$ value at the transition depends on the scheme used: symmetric ($β\sim 0.83$) or non-symmetric ($β\sim 0.61$). Using damage spreading ideas, we also determine the active/chaotic transition line; this line depends on how the replicas are updated.

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