Probability distribution of random walks on self-avoiding walks

The probability distribution of random walks on one-dimensional fractal structures generated by random walks (RW chains) and self-avoiding walks (SAW chains) in d-dimensional space, Pd(r, t), is studied analytically in the case ξ ≡ r/t1/dw ≪ 1, where dw is the fractal dimension of the random walk, 〈r2(t)〉 ∼ t2/dw. It is shown that there exists an infinite hierarchy of critical dimensions, d = dc = 4n + 2, with n ≥ 0 for RW chains and n ≥ 1 for SAW chains, for each term in the ξ-expansion of fd(ξ), the scaling part of Pd(r, t). Each transition is characterized by its own logarithmic correction.