KPZ models: height-gradient fluctuations and the tilt method

TL;DR

This paper investigates the local height-gradient fluctuations in 1D KPZ models, revealing a power-law relationship with discretization step and linking it to interface velocity corrections, enhancing understanding of KPZ universality.

Contribution

It introduces a detailed analysis of the height-gradient fluctuations in discrete KPZ models and connects the fluctuation parameter to finite-size velocity corrections.

Findings

The parameter b approaches 1 as the discretization step increases.

b exhibits a power-law dependence on the step size with an exponent related to the roughness exponent.

Restricted and unrestricted growth models show b approaching 1 from below and above, respectively.

Abstract

When a growing interface belonging to the KPZ universality class is tilted with average slope $m$, its average velocity increases in $\frac{\Lambda}{2}\,m^2$, where $\Lambda$ is related to the nonlinear coefficient $\lambda$ of the KPZ equation. Nevertheless, a necessary condition for this association to hold true is that the mean square height-gradient increases in $b\, m^2$ when the interface is tilted. For the continuous KPZ equation $b = 1$ and the relation $\Lambda=\lambda$ is achieved. In this work, we study the local fluctuations of the height gradient through an analysis of the values of $b$. We show that, for 1-dimensional discrete KPZ models, $b$ has a power-law dependence with the discretization step $s$ chosen to calculate the height gradient and $b$ goes to $1$ as $s$ increases. Its power-law exponent $\gamma_b$ matches the exponent associated with the finite-size corrections of the interface average velocity, $\textit{i.e.}$ $\gamma_b=2(\zeta-1)$, where $\zeta$ is the global roughness exponent. We also show how, for restricted (unrestricted) growth models, the value of $b$ goes to $1$ from below (above) as $s$ increases.