Simplicial complexes: spectrum, homology and random walks

Understand

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.