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.
The high order random walk of REF is designed to expose the topological properties of the complex.
Simplicial complexes: spectrum, homology and random walks
Random Structures&Algorithms, 50(2): 225-261 (2017)
Mathematics Computer Science