Dynamic properties in a family of competitive growing models

HOROWITZ, CLAUDIO M.; ALBANO, EZEQUIEL V.

PHYSICAL REVIEW E - STATISTICAL PHYSICS, PLASMAS, FLUIDS AND RELATED INTERDISCIPLINARY TOPICS

The American Physical Society

Año: 2006 vol. 73 p. 311111 - 311118

1063-651X

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 eposition (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 ($t_{x2}$) 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: $delta = y eta_{RD}$, where $eta_{RD} = 1/2$ is the growing exponent for $RD$. Furthermore, both exponents exhibit universality in the $p ightarrow 0$ 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 $delta$. When the height difference between two neighbouring sites corresponds to a random walk that after walking $langle n angle$ 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 neighbouring sites corresponds to a random walk that after $langle n angle$ steps moves $langle l angle$ 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-MBE and Non-linear-MBE) with the properties of Type A and B of random walks, eight different stochastic equations for all the competitive models studied are derived.