Origins of scaling relations in nonequilibrium growth

Scaling and hyperscaling laws provide exact relations among critical exponents describing the behavior of a system in criticality. For nonequilibrium growth models with a conserved drift there exist few of them: the relation $\alpha +z=4$ thought to be inexact and the laws $2\alpha+d+2=z$ and $2\alpha+d=z$ for conserved and non-conserved noise respectively and attributed to the conserved character of the drift. Herein we show that it is possible to construct conserved surface growth equations for which the relation $\alpha +z=4$ is exact in the renormalization group sense and $2\alpha+d+2=z$ is strongly violated. We explain the presence of scaling and hyperscaling laws in terms of the existence of geometric principles dominating the dynamics.

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