Dynamic properties in a family of competitive growing models

The properties of a wide variety of growing models, generically called X-RD, involving the deposition of particles according to competitive processes, such that a particle is attached to the aggregate with probability p following the mechanisms of a generic model X that provides the correlations and at random [random deposition (RD)] with probability (1-p), 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 (tx2) that, by fixing the sample size, scales with p according to tx2(p) proportional variant p-y (P>0), where is an exponent. Also, the interface width at saturation (Wsat) scales as Wsat(p) proportional variant p-delta (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 --> 0 limit. By mapping the behavior of the average height difference of two neighboring sites in discrete models of type X-RD and two kinds of random walks, we have determined the exact value of the exponent delta. When the height difference between two neighbouring sites corresponds to a random walk that after walking steps returns to a distance from its initial position that is proportional to the maximum distance reached (random walk of type A), one has delta=1/2. On the other hand, when the height difference between two neighboring sites corresponds to a random walk that after steps moves steps towards the initial position (random walk of type B), one has delta=1. Finally, by linking four well-established universality classes (namely Edwards-Wilkinson, Kardar-Parisi-Zhang, linear [molecular beam epitaxy (MBE)] and nonlinear MBE) with the properties of type A and B of random walks, eight different stochastic equations for all the competitive models studied are derived.