Tight inequalities among set hitting times in Markov chains

Given an irreducible discrete-time Markov chain on a finite state space, we consider the largest expected hitting time $T(\alpha)$ of a set of stationary measure at least $\alpha$ for $\alpha\in(0,1)$. We obtain tight inequalities among the values of $T(\alpha)$ for different choices of $\alpha$. One consequence is that $T(\alpha) \le T(1/2)/\alpha$ for all $\alpha < 1/2$. As a corollary we have that, if the chain is lazy in a certain sense as well as reversible, then $T(1/2)$ is equivalent to the chain's mixing time, answering a question of Peres. We furthermore demonstrate that the inequalities we establish give an almost everywhere pointwise limiting characterisation of possible hitting time functions $T(\alpha)$ over the domain $\alpha\in(0,1/2]$.