Width and extremal height distributions of fluctuating interfaces with window boundary conditions

TL;DR

This paper analyzes the statistical distributions of local roughness and extremal heights in fluctuating interfaces across universality classes, revealing scaling laws and universal behaviors useful for experimental and numerical studies.

Contribution

It introduces a scaling framework for squared local roughness and extremal height distributions, providing analytical and numerical evidence of universality across interface classes.

Findings

SLRDs follow log-normal distributions at early times.

Cumulants scale with correlation length and window size.

Universal behavior observed in KPZ class distributions.

Abstract

We present a detailed study of squared local roughness (SLRDs) and local extremal height distributions (LEHDs), calculated in windows of lateral size $l$, for interfaces in several universality classes, in substrate dimensions $d_s = 1$ and $d_s = 2$. We show that their cumulants follow a Family-Vicsek type scaling, and, at early times, when $\xi \ll l$ ($\xi$ is the correlation length), the rescaled SLRDs are given by log-normal distributions, with their $n$th cumulant scaling as $(\xi/l)^{(n-1)d_s}$. This give rise to an interesting temporal scaling for such cumulants $\left\langle w_n \right\rangle_c \sim t^{\gamma_n}$, with $\gamma_n = 2 n \beta + {(n-1)d_s}/{z} = \left[ 2 n + {(n-1)d_s}/{\alpha} \right] \beta$. This scaling is analytically proved for the Edwards-Wilkinson (EW) and Random Deposition interfaces, and numerically confirmed for other classes. In general, it is featured by small corrections and, thus, it yields exponents $\gamma_n$'s (and, consequently, $\alpha$, $\beta$ and $z$) in nice agreement with their respective universality class. Thus, it is an useful framework for numerical and experimental investigations, where it is, usually, hard to estimate the dynamic $z$ and mainly the (global) roughness $\alpha$ exponents. The stationary (for $\xi \gg l$) SLRDs and LEHDs of Kardar-Parisi-Zhang (KPZ) class are also investigated and, for some models, strong finite-size corrections are found. However, we demonstrate that good evidences of their universality can be obtained through successive extrapolations of their cumulant ratios for long times and large $l$'s. We also show that SLRDs and LEHDs are the same for flat and curved KPZ interfaces.