BOND COVERING IN THE LATTICE-COVERING PROBLEM

In this paper, we study the problem of the covering of the bonds of a finite lattice by a random walk that visits all the lattice sites. One-, two-, three-, and four-dimensional regular periodic lattices are considered. While in one-dimension the problem is trivial, our numerical results show very interesting features in higher dimensions, concerning the set of nonvisited bonds when the site covering has been completed. The number of such bonds as a function of the lattice size follows a square-logarithmic law in two dimensions and a power law in three and four dimensions, suggesting the presence of a fractal geometry.