Adhesive riffle shuffles: one-parameter models and mixing bounds

Abstract

We compare four simple shuffles with one adhesion parameter $p$: a card-weighted sticky riffle, a block-weighted riffle, a midpoint-interpolation shuffle, and a correlated-label inverse riffle. Independent surviving contacts create contiguous blocks; no force or energy constants are fitted. We give polynomial upper bounds with explicit dependence on $p$, common adjacency-survival lower bounds, and a sharper information lower bound for interpolation. For fixed deck size $n\ge4$, the mixing time diverges as $Θ((1-p)^{-1})$ for the three cut-first models, but as $Θ((1-p)^{-2})$ for the correlated-label model. The latter distinction comes from whether one or two additional contact breaks are needed to escape cyclic rotations. For even $n$, unglued interpolation retains the lower bound $Ω(\sqrt n\log n)$ and upper bound $O(n^{5/2}\log n)$. We distinguish proved bounds from conjectured scales, recover classical riffle cutoff under sufficiently small perturbations, and explain precisely where established results on clumpy shuffles apply. The fixed-parameter asymptotic orders of the new cut-first families and their possible cutoff remain unresolved.