2012/08/31 by Hyun-Joo Kim, Hyunjoo Kim, Doil Jung +1
Mathematics · Physics and Astronomy · #Approx #Combinatorics #Distribution (mathematics) #Eigenvalues and eigenvectors #Geometry #Lambda #Mathematical analysis #Mathematical physics #Mathematics #Matrix (chemical analysis) #Physics #Quantum mechanics #Random Matrices and Applications #Random matrix #Scaling #Spectral density #Statistics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.stat-mech
paper · pdf · doi:10.1142/s0217984913501972
arxiv created 2012/10/06 · openalex publication_date 2013/10/15 · arxiv updated 2015/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this paper, covariance matrices of heights measured relative to the average height of growing self-affine surfaces in the steady state are investigated in the framework of random matrix theory. We show that the spectral density of the covariance matrix scales as ρ(λ) ~ λ -ν deviating from the prediction of random matrix theory and has a scaling form, ρ(λ, L) = λ -ν f(λ/L ϕ ) for the lateral system size L, where the scaling function f(x) approaches a constant for λ ≪ L ϕ and zero for L ϕ ≪λ< λ max . The values of exponents obtained by numerical simulations are ν ≈ 1.70 and ϕ ≈ 1.51 for the Edward–Wilkinson class and ν ≈ 1.61 and ϕ ≈ 1.76 for the Kardar–Parisi–Zhang class, respectively. The distribution of the largest eigenvalues follows a scaling form as ρ(λ max , L) = 1/L b f max ((λ max - L a )/L b ), which is different from the Tracy–Widom distribution of random matrix theory while the exponents a and b are given by the same values for the two different classes.