Dynamical Inequality in Growth Models

TL;DR

This paper applies a recent exponent inequality to various dynamical growth models, confirming known results, revealing saturation cases, and testing approximation schemes, demonstrating the inequality's utility in analyzing growth exponents.

Contribution

It introduces the application of a new exponent inequality to multiple growth models, assessing the validity of approximation methods and identifying cases where the inequality is saturated.

Findings

Most known exponents are consistent with the inequality.

The Molecular Beam Equation exponents saturate the inequality.

All but one approximation method violate the inequality in some regions.

Abstract

A recent exponent inequality is applied to a number of dynamical growth models. Many of the known exponents for models such as the Kardar-Parisi-Zhang (KPZ) equation are shown to be consistent with the inequality. In some cases, such as the Molecular Beam Equation, the situation is more interesting, where the exponents saturate the inequality. As the acid test for the relative strength of four popular approximation schemes we apply the inequality to the exponents obtained for two Non Local KPZ systems. We find that all methods but one, the Self Consistent Expansion, violate the inequality in some regions of parameter space. To further demonstrate the usefulness of the inequality, we apply it to a specific model, which belongs to a family of models in which the inequality becomes an equality. We thus show that the inequality can easily yield results, which otherwise have to rely either on approximations or general beliefs.