Spectral Density Scaling of Fluctuating Interfaces

Abstract

Covariance matrix of heights measured relative to the average height of a growing self-affine surface 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 $ρ(λ) \sim λ^{-ν}$ 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 $x \ll 1$ and zero for $x \gg 1$. The obtained values of exponents by numerical simulations are $ν\approx 1.73$ and $ϕ\approx 1.40$ for the Edward-Wilkinson class and $ν\approx 1.64$ and $ϕ\approx 1.79$ 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.