Mixing times of one-sided $k$-transposition shuffles

Diagonalizing the transition matrix of a reversible Markov chain is extremely powerful when wanting to prove that the Markov chain exhibits the cutoff phenomenon. The first technique for diagonalizing the transition matrix of a random walk on the Caley graph of a finite group G was introduced by Diaconis and Shahshahani [2]. The technique, which relies on Schur’s lemma, requires understanding of the representation and character theory of G, and has been applied for many random walks on groups [3, 4, 5, 6, 7]. Cases where the generating set is not a conjugacy class are much more challenging. An early example is the case of star transpositions, which was diagonalized by Flatto, Odlyzko and Wales [8]. Diaconis [9] analyzed this diagonalization to show cutoff at n log n. In a recent breakthrough, Dieker and Saliola [1] introduced a new technique to diagonalize the random-to-random shuffle. The proof of cutoff for random-to-random was completed by Bernstein and the first author’s eigenvalue analysis [10] and Subag’s lower bound analysis [11]. Another development was studying the one-sided transposition shuffle on n cards, during which different transpositions are assigned different weights. Bate, Connor and Matheau–Raven [12] diagonalized this shuffle and proved that it exhibits cutoff at n log n. One step of this shuffle consists of choosing a position R uniformly at random, choosing a position L from {1, 2, · · · , R} uniformly at random, and then performing the transposition (RL). Here, we introduce a generalization called the one-sided k−transposition shuffle. As before, we choose a position R uniformly at random, except now, we pick k positions L1, · · · , Lk (not necessarily distinct) uniformly at random from {1, 2, · · · , R}, and perform the permutation (RLk)(RLk−1) · · · (RL1). These products can give rise to many types of permutations of varying weights. Let Pn,k denote the transition matrix of the one-sided k−transposition shuffle on n cards and let U denote the uniform measure on Sn. We define the total variationand l distance between Pn,k and U as follows: