Exact scalings in competitive growth models

TL;DR

This paper derives exact scaling exponents for competitive growth models involving random deposition and other growth rules, revealing how surface roughness scales with probability and identifying transitions between universality classes.

Contribution

It provides exact analytical scaling exponents for specific competitive growth models and explores how their continuous equation coefficients depend on the probability parameter p.

Findings

Exact scaling exponents match previous conjectures.

The coefficients of the continuous equations depend non-trivially on p.

Transition from KPZ to EW universality class as p varies.

Abstract

A competitive growth model (CGM) describes aggregation of a single type of particle under two distinct growth rules with occurrence probabilities $p$ and $1-p$. We explain the origin of scaling behaviors of the resulting surface roughness with respect to $p$ for two CGMs which describe random deposition (RD) competing with ballistic deposition (BD) and RD competing with the Edward Wilkinson (EW) growth rule. Exact scaling exponents are derived and are in agreement with previously conjectured values. Using this analytical result we are able to derive theoretically the scaling behaviors of the coefficients of the continuous equations that describe their universality classes. We also suggest that, in some CGM, the $p-$dependence on the coefficients of the continuous equation that represents the universality class can be non trivial. In some cases the process cannot be represented by a unique universality class. In order to show this we introduce a CGM describing RD competing with a constrained EW (CEW) model. This CGM show a transition in the scaling exponents from RD to a Kardar-Parisi-Zhang behavior when $p \to 0$ and to a Edward Wilkinson one when $p \to 1$. Our simulation results are in excellent agreement with the analytic predictions.