Explicit Cutoff Profiles for Colored Top-$m$-to-Random Shuffles

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 $p$-colored top-$m$-to-random on the wreath product $G_{n,p}=C_p\wr S_n$, with $m$ fixed. Using the Nakano-Sadahiro-Sakurai basis elements $B_m$, we obtain exact nested-set occupancy mixtures and reduce the likelihood ratio to the single statistic $L_p$. This yields exact formulas for separation and $L^\infty(U)$, and exact one-dimensional formulas for total variation, $L^q(U)$ ($1\le q<\infty$), $\chi^2$, and relative entropy. At $k=\Bigl\lfloor \frac{n}{m}(\log n+c)\Bigr\rfloor$, the number of never-chosen labels converges in law to $\mathrm{Poisson}(e^{-c})$, giving the total-variation profile $f_p(c)$, the separation profile, and the corresponding $L^q(U)$, $L^\infty(U)$, $\chi^2$, and relative-entropy profiles. For $m=1$ we recover colored top-to-random; for $p=1$, the total-variation profile reduces to the Diaconis-Fill-Pitman profile.