The Thorp shuffle is defined as follows. Cut a deck of cards into two equal piles. Drop the first card from the left pile or the right pile according to the outcome of a fair coin flip, then drop from the other pile. Continue this way until both piles are empty. We show that the mixing time for the Thorp shuffle with $2^d$ cards is polynomial in d.

The Thorp shuffle is defined as follows. Cut the deck into two equal piles. Drop the first card from the left pile or the right pile according to the outcome of a fair coin flip; then drop from the other pile. Continue this way until both piles are empty. We show that the mixing time for the Thorp shuffle with 2d cards is polynomial in d.

We solve an open problem concerning the mixing time of symmetric random walk on the n-dimensional cube truncated by a hyperplane, showing that it is polynomial in n. As a consequence, we obtain a fully polynomial randomized approximation scheme for counting the feasible solutions of a 0-1 knapsack problem. The results extend to the case of any fixed number of hyperplanes. The key ingredient in our analysis is a combinatorial construction we call a "balanced almost uniform permutation," which is of independent interest.

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.

We solve an open problem concerning the mixing time of a symmetric random walk on an n-dimensional cube truncated by a hyperplane, showing that it is polynomial in n. As a consequence, we obtain a full-polynomial randomized approximation scheme for counting the feasible solutions of a 0-1 knapsack problem. The key ingredient in our analysis is a combinatorial construction we call a "balanced almost uniform permutation", which seems to be of independent interest.