Cutoff profiles for colored top-m-to-random shuffles with growing block size

Abstract

We study the $p$-colored top-$m$-to-random shuffle on $C_p\wr S_n$ when the block size $m=m_n$ grows with $n$. Let $E_{k_n}^{(m_n)}$ be the number of labels never touched after $k_n$ independent uniform $m_n$-subset draws, and set $b_n=n-m_n$, $q_n=b_n/n$, and $λ_n=nq_n^{k_n}$. We prove that if $λ_n\toλ\in(0,\infty)$ and $b_n\to\infty$, then $E_{k_n}^{(m_n)}\Rightarrow\mathrm{Poisson}(λ)$. Combining this with the exact nested-set reduction for colored top-$m$-to-random shuffles, we obtain growing-block total variation, separation, and integrated likelihood-ratio profiles. In particular, if $Q_{n,p}^{(m_n)}$ is the one-step law and $U_{n,p}$ is uniform on $C_p\wr S_n$, then the separation distance from $(Q_{n,p}^{(m_n)})^{*k_n}$ to $U_{n,p}$ tends to $1-e^{-λ}(1+λ)$ for $p=1$ and to $1-e^{-λ}$ for $p\ge2$. The criterion applies to small blocks, proportional blocks, and near-full blocks.