Random walks on finite convex sets of lattice points

This paper examines the convergence of nearest-neighbor random walks on convex subsets of the lattices Z(d). The main result shows that for fixed d, O(gamma(2)) steps are sufficient for a walk to "get random," where gamma is the diameter of the set. Toward this end a new definition of convexity is introduced for subsets of lattices, which has many important properties of the concept of convexity in Euclidean spaces.