A Shuffle that Mixes Sets of any Fixed Size much Faster than it Mixes the Whole Deck

Abstract

Consider an n by n array of cards shuffled in the following manner. An element x of the array is chosen uniformly at random; Then with probability 1/2 the rectangle of cards above and to the left of x is rotated 180 degrees, and with probability 1/2 the rectangle of cards below and to the right of x is rotated 180 degrees. It is shown by an eigenvalue method that the time required to approach the uniform distribution is between n^2/2 and cn^2 ln n for some constant c. On the other hand, for any k it is shown that the time needed to uniformly distribute a set of cards of size k is at most c(k)n, where c(k) is a constant times k^3 ln(k)^2. This is established via coupling; no attempt is made to get a good constant.