We state some results on Cheeger Inequalities for the combinatorial
Laplacian and random walks on simplicial complexes.
Specifically, for the combinatorial Laplacian we prove that a Cheeger
type inequality holds on the highest dimension, or for the boundary
operator with Dirichlet boundary conditions. We also show that
coboundary expanders do not satisfy natural Buser or Cheeger
inequalities. We provide some statements about middle dimensions.
We also introduce random walks with absorbing states on simplicial