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We study the roughening of a growing surface near to a morphological transition within a general scaling framework. For a class of systems, where the transition can be related to directed percolation, the anomalous roughness at the critical point is at most logarithmical. In two-dimensional simulations we find w^2 where w is the width of the surface and L is the substrate size.
Kertész et al. (Mon,) studied this question.