There are intimate connections between the behavior of a random walk on a graph, and its topological and spectral properties. We define a stochastic process on higher dimensional simplicial complexes, which reflects their homological and spectral properties in a parallel way. This leads to high dimensional analogues and counter-analogues of classical theorems of Kesten, Alon-Boppana, and others. No previous knowledge is assumed. Joint work with Ron Rosenthal.