2015/12/16 by Ismael S. S. Carrasco, I. S. S. Carrasco, Tiago J. Oliveira +1
Mathematics · Physics and Astronomy · #Combinatorics #Cumulant #Geometry #Mathematical analysis #Mathematical physics #Mathematics #Physics #Quantum mechanics #Random Matrices and Applications #Renormalization group #Scaling #Statistical physics #Statistics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Universality (dynamical systems) #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.93.012801
published as Phys. Rev. E 93, 012801 (2016) · 11 pages, 10 figures, 4 tables
arxiv created 2015/12/16 · openalex publication_date 2016/01/07 · arxiv updated 2016/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 ds=1 and 2. We show that their cumulants follow a Family-Vicsek-type scaling, and, at early times, when \ensuremathξ\ensuremath≪l (\ensuremathξ is the correlation length), the rescaled SLRDs are given by log-normal distributions, with their nth cumulant scaling as (\ensuremathξ/l)^(n\ensuremath-1)ds. This gives rise to an interesting temporal scaling for such cumulants as \ensuremath⟨wn\ensuremath⟩c\ensuremath∼t^\ensuremathγn, with \ensuremathγn=2n\ensuremathβ+(n\ensuremath-1)ds/z=[2n+(n\ensuremath-1)ds/\ensuremathα]\ensuremathβ. 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 \ensuremathγn (and, consequently, \ensuremathα,\ensuremathβ and z) in good agreement with their respective universality class. Thus, it is a useful framework for numerical and experimental investigations, where it is usually hard to estimate the dynamic z and mainly the (global) roughness \ensuremathα exponents. The stationary (for \ensuremathξ\ensuremath≫l) SLRDs and LEHDs of the Kardar-Parisi-Zhang (KPZ) class are also investigated, and, for some models, strong finite-size corrections are found. However, we demonstrate that good evidence of their universality can be obtained through successive extrapolations of their cumulant ratios for long times and large l. We also show that SLRDs and LEHDs are the same for flat and curved KPZ interfaces.