Weierstrassian Levy flights and self-avoiding random walks

An analogy, involving space–time relationships, is made between a self-avoiding random walk and a Weierstrass random walk. The self-avoiding random walk is non-Markovian while the Weierstrass random walk is Markovian, but with a built-in self-similarity characterized by a fractal dimension. It is proposed that the self-avoiding random walk can be viewed as a Weierstrass random walk whose trajectory is composed of maximally packed self-similar clusters with the symmetry of the underlying lattice.