A new kinetic walk and percolation perimeters.

Abstract

We introduce the smart kinetic walk (SKW), a new kinetic-walk model which is in a different universality class from other such models. The SKW is strictly self-avoiding, yet is never forced to terminate, because it never starts down a path which would lead to its being trapped. We show that a ring-forming version of the model in two dimensions traces out the external perimeter of critical percolation clusters. Using previous results on these perimeters, we find that the SKW fractal dimension is ${D}_{\mathrm{SKW}\mathrm{\ensuremath{\approxeq}}}$1.75. The equivalence between the walk and percolation perimeters leads to a scaling form for the number of N-step rings. Finally, we see that the walk with a bias to turn to the left more often than to the right (or vice versa) traces out the perimeter of clusters that are not at the percolation threshold. This implies a maximum ring size depending upon the strength of the bias.