TITLE: Minimum number of random moves needed to uniformly scramble a Rubik's cube?
QUESTION [5 upvotes]: Follow up from my last question: $3 \times 3$ Rubik's cube scrambling question
I am talking about $3 \times 3$ Rubik's cubes. Start with a solved cube. Then make some amount of random moves (where moves are defined using the half-turn metric: any twist of the face, i.e. 90 degrees counterclockwise, 90 degrees clockwise, 180 degrees are each one move). After how many moves will each of the 43 quintillion states be equally likely? If the answer is "infinitely many," can someone give some idea of how many moves will be "close enough?"

REPLY [2 votes]: You only converges towards the uniform distribution, so you a priori need an infinite number of shuffles...The good concept, however, is the one of "Markov chain mixing time", i.e., the number of step so that any initial Dirac distribution is transformed into a distribution at distance at most 1/4 (this is an arbitrary value) from the uniform distribution. A very good book on the subject : Markov chain mixing time, by Yuval Peres, Elizabeth Wilmer and David Levin. There are also interesting results in "Random Walks on Graphs: A Survey" by Laslo Lovasz.
You can easily show that the mixing time of the Rubik cube is at least 18, simply using the total number of configurations and the fact that you can reach at most 12 new cubes after each  move. It is rather close to the diameter of the Rubik graph (20) or the average distance between two cubes (around 17).
General theorems yield poor upper bound (I remember having computed some time ago the upper bound 1600 moves). My personnal guess is that something like 45 moves should be sufficient, but I have no proof. Actually, it is mainly based on my experiments on the 2x2x2 cube (pocket cube) : in this case I was able to compute exhaustively how the distance to the uniform distribution varies: I found that the mixing time is of 24 or 25 moves (24 or 25 because the graph is bipartite). Recall that the diameter is 13 in this case.
I would be very interested in further results. Can we for example approximate the expander ratio of the Rubik cube graph? The second largest eigenvalue of the adjacency matrix?