Width and extremal height distributions of Kardar-Parisi-Zhang interfaces with window boundary conditions.

We present a detailed numerical study of squared local roughness (SLRDs) and local extremal height distributions (LEHDs), calculated in windows of lateral size $l$, for KPZ interfaces, in $1+1$ and $2+1$ dimensions. We show that their cumulants follow the Family-Vicsek type scaling, and, at early times (when the correlation length $\xi$ is smaller than $l$), the $n$th SLRDs cumulant scale in time as $\left\langle w_n \right\rangle_c \sim t^{\gamma_n}$, with $\gamma_n = 2 n \beta + {(n-1)d_s}/{z} = \left[ 2 n + {(n-1)d_s}/{\alpha} \right] \beta$. This scaling is featured by small corrections, providing exponents ($\alpha$, $\beta$ and $z$) in nice agreement with their respective universality classes. Therefore, it is an useful framework for numerical and experimental investigations, where it is, usually, hard to estimate the dynamic $z$ and the (global) roughness $\alpha$ exponents. At the stationary regime ($\xi \gg l$), we observe that SLRDs and LEHDs for small $l$'s can present strong finite-size corrections, and the best way to prove their universality is through extrapolations (for long times and $l$'s) of their cumulant ratios. This shows that the procedure adopted in recent comparisons of experimental and numerical/theoretical distributions, based on small $l$ data collapse, could lead to misleading conclusions. As final result, we demonstrate that SLRDs and LEHDs are the same for flat interfaces and curved ones. We acknowledge FAPEMIG, CAPES and CNPq by the financial support.