The crumpled state of some non-equilibrium fractal surfaces

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Abstract
The authors 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 they find that (1) the width of P(x, r) 'diverges' for r near the ensemble average radius R of the CS and (2) (x) approximately r13.