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.
We show that the expansion of the links implies the concentration of the spectrum of the high order normalized adjacency matrices, a thing that can not be deduced in any way from REF .
Simplicial complexes: spectrum, homology and random walks
Random Structures&Algorithms, 50(2): 225-261 (2017)
Mathematics Computer Science