Probability distribution of random walks on self-avoiding walks

Abstract

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, , is studied analytically in the case , where is the fractal dimension of the random walk, . It is shown that there exists an infinite hierarchy of critical dimensions, , with for RW chains and for SAW chains, for each term in the -expansion of , the scaling part of . Each transition is characterized by its own logarithmic correction.