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