Origins of scaling relations in nonequilibrium growth

TL;DR

This paper investigates the origins of scaling relations in nonequilibrium surface growth, demonstrating that certain models can exactly satisfy the relation α + z = 4 through geometric principles, with implications for understanding critical behavior.

Contribution

It constructs conserved surface growth equations in 2D where the relation α + z = 4 is exact, clarifying the geometric basis of this scaling law.

Findings

Exact α + z = 4 relation in specific models

Geometric principles underpin the scaling law

Renormalization group confirms the relation's validity

Abstract

Scaling and hyperscaling laws provide exact relations among critical exponents describing the behavior of a system at criticality. For nonequilibrium growth models with a conserved drift there exist few of them. One such relation is $\alpha +z=4$, found to be inexact in a renormalization group calculation for several classical models in this field. Herein we focus on the two-dimensional case and 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. We explain the presence of this scaling law in terms of the existence of geometric principles dominating the dynamics.