Topological properties of percolation clusters

Abstract

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 Lv. 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.