Growth Perimeters Generated by a Kinetic Walk: Butterflies, Ants and Caterpillars

Abstract

Diffusion processes can be represented by a random walk on a fractal substrate. One might also study a walk that creates its own substrate by certain rules while walking: The trace left by such a walk forms a cluster of visited sites S. E.g., a walker on a lattice, which at each step chooses its direction at random and occupies the next site with probability p_c (or blocks it with probability l − p_c), creates a percolation cluster at the percolation threshold p_c. How does this cluster grow in time? The growth occurs, apparently, through the sites that are nearest neighbors to visited sites but which were not tested before by the walk (not yet blocked or visited). These sites are called growth sites¹ and the set of these sites we call a growth perimeter G. One can appreciate that the direct study of growth perimeters leads to an interesting problem that has two aspects: pure growth patterns and the types of diffusion leading to these patterns. Here we will discuss a new butterfly walk² that visits only the growth sites and, therefore, concentrates on a pure growth phenomena. In the end we will return to the diffusion aspect of the problem and compare butterfly diffusion to the random diffusion on percolation (the “ant”).