A lower bound for the mixing time of the random-to-random insertions shuffle

Abstract

The best known lower and upper bounds on the total variation mixing time for the random-to-random insertions shuffle are (1/2- o (1))n log n and (2 + o (1)) n log n. A long standing open problem is to prove that the mixing time exhibits a cutoff. In particular, Diaconis conjectured that the cutoff occurs at 3 4n log n. Our main result is a lower bound of tn = (3/4 - o (1))n log n, corresponding to this conjecture. Our method is based on analysis of the positions of cards yet-to-be-removed. We show that for large n and tn as above, there exists f(n) = Θ(√n log n)such that, with high probability, under both the measure induced by the shuffle and the stationary measure, the number of cards within a certain distance from their initial position is f(n) plus a lower order term. However, under the induced measure, this lower order term is strongly influenced by the number of cards yet-to-be-removed, and is of higher order than for the stationary measure.