Gaps: the Strategy behind a Speedy Coupling of Card Shuffling

Using methods of coupling we analyze two models of card shuffling, the random transposition and the random-to-random process, to find minimal coupling times. Disregarding the rank and suit on the standard playing cards we are familiar with, we focus only on the color; specifically, we consider decks where half the cards are white and half are black. We define the total gaps on a pair of decks, which proves to be a valuable distance and technique to a quick coupling. Using the total gaps and taking advantage of the color scheme of the decks, we bound the mixing time of the random-to-random shuffle to O(n logn). Along the way, we prove useful properties unique to a deck with only two colors.