I heard about this cool-sounding seminar at the Berkeley Statistics department:

Mixing time of the card-cyclic to random shuffle
Ben Morris

We analyze the following method for shuffling cards. First, remove card 1 (i.e., the card with label 1) and then re-insert it randomly into the deck. Then repeat with cards 2, 3,…, . Call this a round. R. Pinsky showed, somewhat surprisingly, that the mixing time is greater than one round. We show that in fact the mixing time is on the order of rounds.

The talk is based on a paper with Weiyang Ning (a student at UW) and Yuval Peres. The description the results is somewhat different because it’s rounds of moves, or moves. From the intro to the paper:

To prove the lower bound we introduce the concept of a barrier between two parts of the deck that moves along with the cards as the shuffling is performed. Then we show that the trajectory of this barrier can be well-approximated by a deterministic function … we relate the mixing rate of the chain to the rate at which converges to a constant.

The key is to use path coupling, a technique pioneered by Bubley and Dyer. It’s a cute result and would be a fun paper to read for a class project or something, I bet.