A growing self-avoiding walk in three dimensions and its relation to percolation

Abstract

We introduce a growing self-avoiding walk in three dimensions (3D) that can terminate only by returning to its point of origin. This tricolor walk depends on two parameters, p and q, and is a direct generalization of the smart kinetic walk to 3D. Our walk is closely related to percolation with three colors (black, white, and gray): the tricolor walk directly constructs a loop formed by the confluence of a black, a white, and a gray cluster. The parameters p and q are the fraction of sites colored black and white, respectively. We present numerical and analytical evidence that for p=q=1/3, the fractal dimension of the tricolor walk is exactly 2. For p=q<1/3, the walks undergo a percolation transition at p0.2915. Our Monte Carlo simulations strongly suggest that this transition is not in the same universality class as the usual percolation transition in 3D. The mean length of the finite walks is divergent throughout an extended region of the parameter space. © 1992 The American Physical Society.