Sampling-induced hidden cycles in correlated random rough surfaces

We show both experimentally and theoretically that the sampling-induced hidden cycles can exist in scale-invariant rough surfaces having a correlation length ξ. If the sampling size L is sufficiently large, the oscillatory behavior will diminish with the fluctuation within an order of (ξ/L)^d/2. This is consistent with the law of large numbers for the correlated systems: the average of N-correlated variables having a correlation length ξ will converge to their mean within an order of √ξ^d/N. Based on this result, we propose that in order to distinguish the mound surface from the self-affine surface, the sampling condition √ξ^d/N≪1 and an average of a large number of images are required.