Explicit Cutoff Profiles for Colored Top-$m$-to-Random Shuffles
Ivan Z. Feng

TL;DR
This paper derives exact formulas for cutoff profiles and various distance measures for colored top-$m$-to-random shuffles on wreath products, generalizing known results and providing precise asymptotics.
Contribution
It introduces explicit cutoff profiles for colored top-$m$-to-random shuffles on wreath products, extending classical results and providing exact formulas for multiple distance metrics.
Findings
Exact formulas for separation, total variation, $oldsymbol{L^q}$, $oldsymbol{ ext{chi}^2}$, and relative entropy profiles.
Poisson convergence of never-chosen labels at the cutoff point.
Recovery of known profiles for special cases like uncolored top-to-random and classical top-to-random.
Abstract
We study -colored top--to-random on the wreath product , with fixed. Using the Nakano-Sadahiro-Sakurai basis elements , we obtain exact nested-set occupancy mixtures and reduce the likelihood ratio to the single statistic . This yields exact formulas for separation and , and exact one-dimensional formulas for total variation, (), , and relative entropy. At , the number of never-chosen labels converges in law to , giving the total-variation profile , the separation profile, and the corresponding , , , and relative-entropy profiles. For we recover colored top-to-random; for , the total-variation profile reduces to the Diaconis-Fill-Pitman profile.
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