Fractal Properties of “Disordered Surfaces” and The Termite Problem

Abstract

In this second lecture, I shall turn to the topic of fractal surfaces, for which knowledge of only “the” fractal dimension d_f is not sufficient to describe the essential physics. In fact, we must distinguish between the entire external perimeter or “hull,” with its corresponding fractal dimension d_h, and the unscreened portion of the external perimeter and its fractal dimension d. A very simple argument gives rise to an expression for d_u in terms of d_f and d_p, the fractal dimension of the particles used to probe the surface. We shell see that the concept of the unscreened perimeter is at the heart of transport in a two-component random medium made up of superconductors and normal resistors. In particular, the exponent $\tilde s = s/v$ for characterizing the divergence of the macroscopic conductivity at the percolation threshold p_c is identically equal to d_u.