Combinatorica
SIAM Journal on Computing
Algorithms for random generation and counting: a Markov chain approach
Algorithms for random generation and counting: a Markov chain approach
Faster mixing via average conductance
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Edge isoperimetry and rapid mixing on matroids and geometric Markov chains
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Blocking Conductance and Mixing in Random Walks
Combinatorics, Probability and Computing
Finding sparse cuts locally using evolving sets
Proceedings of the forty-first annual ACM symposium on Theory of computing
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We show that a new probabilistic technique, recently introduced by the first author, yields the sharpest bounds obtained to date on mixing times 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 |pn(x,y)/π(y)-1| of the distribution at time n from the stationary distribution π. Our approach also yields a direct link between isoperimetric inequalities and heat kernel bounds; previously, this link rested on analytic estimates known as Nash inequalities.