Abstract
We study the time dependence of the auto-correlation function of surface fluctuations in randomly growing clusters in a radial geometry. We show that large-scale surface fluctuations have qualitatively different behaviour than in the cylindrical geometry where the transverse size of the system remains constant. For a class of Langevin dynamics characterized by the dynamic exponent z, we calculate the height–height auto-correlation function analytically and show that it tends to a non-zero value in the long time limit, unlike in the cylindrical geometry, where the auto-correlation function decays to zero. However, the persistence probability that surface fluctuation in a given direction does not change sign decays to zero at large times following a power law. We determine the persistence exponent θ numerically, and also analytically approximately by using a perturbation theory. The exponent θ is found to be weakly dependent on z. We propose an approximate phenomenological description for the time dependence of the auto-correlation function in radial growth of surfaces described by KPZ equation, which is in good agreement with our simulations of growing Eden clusters in two dimensions. We argue that the auto-correlation exponent λ and the growth exponent β for growing rough surfaces in (d+1)-dimensional hyperspherical geometry satisfy the relation λ = β.