Cutoff and a Gaussian-shift profile for the Rudvalis shuffle

Abstract

The Rudvalis shuffle removes the top card and reinserts it uniformly at random in one of the bottom two positions. Hildebrand proved the classical upper bound $O(n^3\log n)$, and Wilson later proved the sharp lower coefficient $1/(8π^2)$, establishing the correct order but leaving the leading upper constant, cutoff, and the cutoff profile open. We prove that the full labelled shuffle has total-variation cutoff at \[ \frac{n^3\log n}{8π^2} \] with a window of order $n^3$. The upper bound is obtained by writing one deterministic rotation of the shuffle as a nonnormal sweep of adjacent averaging projections, comparing that sweep to a weighted interchange generator, and controlling the entire regular-representation heat trace. The key sharp-window ingredients are heat-kernel smoothing through the Octopus Inequality, zero extension of deletion-harmonic ordered-tuple functions, a representation-theoretic multiplicity payment, and a sectorwise entrance estimate that disposes of all large Young-diagram deficits. For the profile, an exact complex one-card eigenfunction yields a deck statistic whose surviving first Fourier mode converges to a Gaussian shift. A matching full-deck $L^2$ bound then shows that this one-dimensional Gaussian likelihood ratio asymptotically exhausts all remaining information, giving the exact total-variation profile.