Improved bounds for the mixing time of the random-to-random insertion   shuffle
  We prove an upper bound of $1.5324 n \log n$ for the mixing time of the random-to-random insertion shuffle, improving on the best known upper bound of $2 n \log n$. Our proof is based on the analysis of a non-Markovian coupling. 
Ben Morris, Chuan Qin
math.PR