  We prove a theorem that reduces bounding the mixing time of a card shuffle to
verifying a condition that involves only pairs of cards, then we use it to
obtain improved bounds for two previously studied models.
  E. Thorp introduced the following card shuffling model in 1973. Suppose the
number of cards n is even. Cut the deck into two equal piles. Drop the first
card from the left pile or from 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 obtain a mixing time bound of O(log^4 n). Previously, the
best known bound was O(log^{29} n) and previous proofs were only valid for n a
power of 2.
  We also analyze the following model, called the L-reversal chain, introduced
by Durrett. There are n cards arrayed in a circle. Each step, an interval of
cards of length at most L is chosen uniformly at random and its order is
reversed. Durrett has conjectured that the mixing time is O(max(n, n^3/L^3) log
n). We obtain a bound that is within a factor O(log^2 n) of this,the first
bound within a poly log factor of the conjecture.
