The Range of a Simple Random Walk on Z: An Elementary Combinatorial Approach

Two different elementary approaches for deriving an explicit formula for the distribution of the range of a simple random walk on Z of length n are presented. Both of them rely on Hermann Weyl’s discrepancy norm, which equals the maximal partial sum of the elements of a sequence. By this the original combinatorial problem on Z can be turned into a known path-enumeration problem on a bounded lattice. The solution is provided by means of the adjacency matrix Qd of the walk on a bounded lattice (0, 1, . . . , d). The second approach is algebraic in nature, and starts with the adjacency matrix Qd. The powers of the adjacency matrix are expanded in terms of products of non-commutative left and right shift matrices. The representation of such products by means of the discrepancy norm reveals the solution directly.