Properties of the skeleton of aggregates grown on a Cayley tree

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_l^s, can be expressed in terms of d_l as d_l^s = 1, d_l≤ d_l^c = 2;d_l^s = d_l -1, d₁ ≥d_l^c = 2, which implies that the tree is finitely ramified for d_l≤ 2 and infinitely ramified when d_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.