KPZ models: height-gradient fluctuations and the tilt method

When a growing interface belonging to the KPZ universality class is tilted with average slope $m$, its average velocity increases in $\frac{\mathrm{\Lambda }}{2}{m}^2$, where $\mathrm{\Lambda }$ 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 $\mathrm{\Lambda }=\lambda$ 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, $\mathit{\text{i.e.}}$ ${\gamma }_b=2(\zeta -1)$, 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.