Dynamic properties in a family of competitive growing models

The properties of a wide variety of growing models, generically called \(X/RD\), are studied by means of numerical simulations and analytic developments. The study comprises the following \(X\) models: Ballistic Deposition, Random Deposition with Surface Relaxation, Das Sarma-Tamboronea, Kim-Kosterlitz, Lai-Das Sarma, Wolf-Villain, Large Curvature, and three additional models that are variants of the Ballistic Deposition model. It is shown that after a growing regime, the interface width becomes saturated at a crossover time (\(t_{x2}\)) that, by fixing the sample size, scales with \(p\) according to \(t_{x2}(p)\propto p^{-y}, \qquad (p > 0)\), where \(y\) is an exponent. Also, the interface width at saturation (\(W_{sat}\)) scales as \(W_{sat}(p)\propto p^{-\delta}, \qquad (p > 0)\), where \(\delta\) is another exponent. It is proved that, in any dimension, the exponents \(\delta\) and \(y\) obey the following relationship: \(\delta = y \beta_{RD}\), where \(\beta_{RD} = 1/2\) is the growing exponent for \(RD\). Furthermore, both exponents exhibit universality in the \(p \to 0\) limit. By mapping the behaviour of the average height difference of two neighbouring sites in discrete models of type \(X/RD\) and two kinds of random walks, we have determined the exact value of the exponent \(\delta\). Finally, by linking four well-established universality classes (namely Edwards-Wilkinson, Kardar-Parisi-Zhang, Linear-MBE and Non-linear-MBE) with the properties of both random walks, eight different stochastic equations for all the competitive models studied are derived.