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

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 $b_n=n-m_n$, $q_n=b_n\mathrm{/}n$, and ${\lambda }_n=n{q}_n^{k_n}$. We prove that if ${\lambda }_n\to \lambda \in (0,\mathrm{\infty })$ and $b_n\to \mathrm{\infty }$, then $E_{k_n}^{(m_n)}\Rightarrow \mathrm{P}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{n}(\lambda )$. 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 $(Q_{n,p}^{(m_n)}{)}^{\ast k_n}$ to $U_{n,p}$ tends to $1-e^{-\lambda }(1+\lambda )$ for $p=1$ and to $1-e^{-\lambda }$ for $p\ge 2$. The criterion applies to small blocks, proportional blocks, and near-full blocks.