  We show that a new probabilistic technique, recently introduced by the first
author, yields the sharpest bounds obtained to date on mixing times of Markov
chains in terms of isoperimetric properties of the state space (also known as
conductance bounds or Cheeger inequalities). We prove that the bounds for
mixing time in total variation obtained by Lovasz and Kannan, can be refined to
apply to the maximum relative deviation $|p^n(x,y)/\pi(y)-1|$ of the
distribution at time $n$ from the stationary distribution $\pi$. We then extend
our results to Markov chains on infinite state spaces and to continuous-time
chains. Our approach yields a direct link between isoperimetric inequalities
and heat kernel bounds; previously, this link rested on analytic estimates
known as Nash inequalities.
