THE CRUMPLED STATE OF SOME NONEQUILIBRIUM FRACTAL SURFACES

We study how the three-dimensional `air' or Pythagorean distance r(Q,Q') between two points Q and Q' on a non-equilibrium crumpled fractal surface (CS), with the topology of the plane, transforms in the internal or geodesic distance x(Q, Q')-with probability P(x, r)-after the unfolding of the CS on a plane. The probability distribution P(x, r) governing this process is examined for the first time. Among other results we find that (1) the width of P(x, r) `diverges' for r near the ensemble average radius R of th CS and (2) [x] approximately r 1/3.