The autocorrelation function for island areas on self-affine surfaces

The spatial distribution of regions that lie above contours of constant height through a self-affine surface is studied as a function of the Hurst exponent H. If the surface represents a landscape, these regions correspond to islands. When the surface represents the height difference for contacting surfaces, the regions correspond to mechanical contacts in the common bearing area model. The autocorrelation function C(Δr) is defined as the probability that points separated by Δr are both within islands. The scaling of C has important implications for the stiffness and conductance of mechanical contacts. We find that its Fourier transform C(q) scales as a power of the wavevector magnitude q: [Formula: see text] with μ = 2 + H rather than the value μ = 2 + 2H reported previously. An analytic argument for μ = 2 + H is presented using the distribution of areas contained in disconnected islands.