Given a simplicial complex X, the k-dimensional random walk on it is the random walk between k-faces of X through (k+1)-faces. We show that under suitable assumptions on the spectral gaps of the links of the X, we can bound (or even determine up to a small error) the rate of convergence of the random walk for a given function on k-faces.

This result fits into a larger framework, in which we think about a simplicial complex with large spectral gaps in the links as a (spectral) high dimensional analogue of an expander graph.

This talk is based on a joint work with Tali Kaufman.