Properties of the skeleton of aggregates grown on a Cayley tree

Abstract

We discuss and analyze a family of trees grown on a Cayley tree, that allows for a variable exponent in the expression for the mass as a function of chemical distance, 〈M(l)〉∼l ^dl. For the suggested model, the corresponding exponent for the mass of the skeleton,d ^s _l , can be expressed in terms ofd _l asd ^s _l = 1,d _l ⩽ d ^c _l = 2;d ^s _l = d _l −1,d ₁ ⩾d ^c _l = 2, which implies that the tree is finitely ramified ford _l ⩽ 2 and infinitely ramified whend _l ⩾ 2. Our results are derived using a recursion relation that takes advantage of the one-dimensional nature of the problem. We also present results for the diffusion exponents and probability of return to the origin of a random walk on these trees.