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 xi. If the sampling size L is sufficiently large, the oscillatory behavior will diminish with the fluctuation within an order of (xi/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 xi will converge to their mean within an order of root xi(d)/N. Based on this result, we propose that in order to distinguish the mound surface from the self-affine surface, the sampling condition root xi(d)/N much less than 1 and an average of a large number of images are required.