Lumping and the mixing time of a reversible chain: the dependence on the number of states

Aldous and Fill asked whether the stopping-rule mixing time $\tau_1^{(2)}$ of a chain obtained from a reversible Markov chain by lumping states is at most an absolute constant times that of the original chain. Hermon and Peres answered the question in the negative: for random walks on graphs of bounded degree, lumping can increase the mixing time by a factor of the order of the logarithm of the number of vertices, and by no more. We consider general reversible chains with $n$ states and show that the largest possible factor $K(n)$ satisfies $\lfloor (n-2)/2\rfloor\le K(n)\le n-1$. The upper bound follows from standard inequalities and from the inequality $\tau_2\le\tau_1^{(2)}$ between the relaxation time and the stopping-rule mixing time, for which we give a short proof; the lower bound comes from an explicit random walk on a weighted comb. We also show that lumping increases the mean hitting time of a set $A$ by a factor of at most $n-|A|$, and that this bound cannot be improved by more than a factor of about two.