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

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 p≃0.2915. 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 χ is divergent throughout an extended region of the parameter space.