Universal scaling relations for growth phenomena

The Family–Vicsek (FV) relation is a seminal universal relation obtained for the global roughness at the interface of two media in the growth process. In this work, we revisit the scaling analysis and, through both analytical and computational means, show that the FV relation can be generalized to a new scaling independent of the size, substrate dimension d , and scaling exponents. We use the properties of lattice growth models in the Kardar–Parisi–Zhang and Villain–Lai–Das Sarma universality classes for 1 ⩽ d ⩽ 3 to support our claims.