In Chapter 4 yyy we discussed equivalences between different definitions of “mixing time” in the family. Lovasz and Winkler [240] give a detailed treatment of analogous results in the non-reversible case.
xxx state some of this ?
Any Markov chain can be viewed as random walk on a weighted directed graph, but even on unweighted digraphs it is hard to relate properties on the walk to graph-theoretic properties, because (as we have often observed) it is in general hard to get useful information about the stationary distribution. An exception is the case of a balanced digraph, i.e. when the in-degree equals the out-degree (, say) at each vertex . Random walk on a balanced digraph clearly retains the “undirected” property that the stationary probabilities are proportional to . Now the proofs of Theorems yyy and yyy in Chapter 6 extend unchanged to the balanced digraph setting, showing that the cover-and-return time satisfies
(The proofs rely on the edge-commute inequality (Chapter 3 yyy), rather than any “resistance” property).
Consider a Markov chain on states for which the only possible transitions are downward, i.e. for we have
and . The chain is ultimately absorbed in state . A question posed by Gil Kalai is whether there is a bound on the mean absorption time involving a parameter similar to that appearing in Cheeger’s inequality. For each proper subset of with define
and then define
Prove that is bounded by a polynomial function of .