Local spectral expansion is a very useful method for arguing about the spectral properties of several random walk matrices over simplicial complexes. The motivation of this work is to extend this method to analyze the mixing times of Markov chains for combinatorial problems. Our main result is a sharp upper bound on the second eigenvalue of the down-up walk on a pure simplicial complex, in terms of the second eigenvalues of its links. We show some applications of this result in analyzing mixing times of Markov chains including sampling independent sets of a graph.

(https://arxiv.org/abs/2001.02827)
Joint work with: Lap Chi Lau