Review on Relationship Between the Universality Class of the Kardar-Parisi-Zhang Equation and the Ballistic Deposition Model

Abstract

We have analysed the research findings on the universality class and discussed the connection between the Kardar-Parisi-Zhang (KPZ) universality class and the ballistic deposition model in microscopic rules. In one dimension and 1+1 dimensions deviations are not important in the presence of noise. At the same time, they are very relevant for higher dimensions or deterministic evolution. Mostly, in the analyses a correction scale higher than 1280 has not been studied yet. Therefore, the growth of the interface for finite system size β ≥ <i>0.30</i> value predicted by the KPZ universality class is still predominant. Also, values of α ≥ <i>0.40,</i> β ≥ <i>0.30,</i> and <i>z</i> ≥ <i>1.16</i> obtained from literature are consistent with the expected KPZ values of α = <i>1</i>/<i>2</i>, β = <i>1</i>/<i>3</i>, and <i>z</i> = <i>3</i>/<i>2</i>. A connection between the ballistic deposition and the KPZ equation through the limiting procedure and by applying the perturbation method was also presented.