  The motivation of this work is to extend the techniques of higher order
random walks on simplicial complexes to analyze 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 and sampling common independent sets of two partition matroids.
