[Pemantle 1989](https://www.math.upenn.edu/~pemantle/papers/overhand2.pdf) proves, among other things, that the Markov chain on $S\_n$ induced by repeatedly and independently performing an [overhand shuffle](https://en.wikipedia.org/wiki/Shuffling#Overhand_shuffle) on a deck of $n$ cards is ergodic and has limiting distribution $U$, the uniform distribution on $S\_n$.


What bothers me is that this result is announced without much fanfare, in fact it's never even made explicit: it's an implicit consequence of Theorem 1 in that paper. Pemantle simply begins by assuming that the limit distribution is $U$ and then produces bounds to prove it. But to me this is a very surprising result. When we perform an overhand shuffle, we randomly select one of a certain subset $O$ of permutations on the deck, and it's not even obvious to me that the subgroup generated by $O$ is equal to $S\_n$. That the limiting distribution is $U$ is equivalent to the transition matrix being double stochastic, which is not obvious to me either.


My question is: is there some more general result about random walks on finite groups or on $S\_n$ which predicts that the limiting distribution is $U$? Or alternatively, is there some simple argument I'm missing to show that the transition matrix in this case is double stochastic?



Why does the overhand shuffle converge to the uniform distribution on $S_n$?
Shuffles like the overhand shuffle or riffle shuffle are not just random walks, they are symmetric random walks because you apply a random permutation drawn from the same distribution no matter what the initial configuration is. You can apply a shuffle with the cards face down. (That's not the case for non-shuffle random walks like "Deal out the first 5-10 cards and sort them.") The transition matrices are automatically doubly stochastic. They preserve the uniform distribution. 


As long as there is some $t$ so that after $t$ shuffles, all permutations occur with positive probability (the shuffles generate $S\_n$ and not all shuffles are odd), then the Perron-Frobenius theorem says that the limiting distribution is uniform, and the question is how long it takes to get close to uniform by various metrics. 


