  Random walks on a graph reflect many of its topological and spectral
properties, such as connectedness, bipartiteness and spectral gap magnitude. In
the first part of this paper we define a stochastic process on simplicial
complexes of arbitrary dimension, which reflects in an analogue way the
existence of higher dimensional homology, and the magnitude of the
high-dimensional spectral gap originating in the works of Eckmann and Garland.
  The second part of the paper is devoted to infinite complexes. We present a
generalization of Kesten's result on the spectrum of regular trees, and of the
connection between return probabilities and spectral radius. We study the
analogue of the Alon-Boppana theorem on spectral gaps, and exhibit a
counterexample for its high-dimensional counterpart. We show, however, that
under some assumptions the theorem does hold - for example, if the
codimension-one skeletons of the complexes in question form a family of
expanders.
  Our study suggests natural generalizations of many concepts from graph
theory, such as amenability, recurrence/transience, and bipartiteness. We
present some observations regarding these ideas, and several open questions.
