Cutoff profiles for conjugacy invariant random walks on symmetric groups

Abstract

We prove asymptotic equivalents for finite-level representations of symmetric groups, that is, for Young diagrams having all but finitely many boxes on their first row. We deduce that random walks on symmetric groups generated by conjugacy classes with a macroscopic number of fixed points have a Poissonian cutoff profile. We also prove that the random involution walk exhibits cutoff and find its cutoff profile. Finally, we obtain numerics for the random transposition walk on a deck of 52 cards, giving concrete estimates on the question that originally motivated Diaconis and Shahshahani.