Topological properties of percolation clusters

Percolation properties, including the total cluster mass S, the shell mass B, and the linear geometrical size R, are studied as a function of the topological 'chemical distance' parameter L. All critical exponents are shown to be related to an apparently new exponent nu , defined by R approximately L ^v. Critical exponents are calculated exactly for percolation clusters on the Cayley tree (a model for 6D percolation), for which S approximately L², B approximately L, and R² approximately L. For diffusion on such clusters one finds that L³ approximately t. Numerical estimates of the exponents are obtained for other dimensions. A conjecture which relates nu to beta and nu is discussed.