Séminaire Lotharingien de Combinatoire, 93B.6 (2025), 11 pp.
Evita Nestoridi and Alan Yan
Cutoff for the Biased Random Transposition Shuffle
Abstract.
We study the biased random transposition shuffle, a natural
generalization of the classical random transposition shuffle studied
by Diaconis and Shahshahani. Using the representation theory of the
symmetric group, we diagonalize the transition matrix of the
shuffle. We use these eigenvalues to prove that the shuffle exhibits
total variation cutoff at time tN = (1/2b) N log N with
window N. We also prove that the limiting distribution of the number
of fixed cards near the cutoff time is Poisson.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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