Existence of the random-to-random shuffle limit profile

Determine whether the total variation limit profile exists for the random-to-random card shuffle at its cutoff time and window.

Background

The random-to-random card shuffle selects a card uniformly, removes it, and reinserts it at a uniformly chosen position. The paper states that this shuffle has cutoff with a specified cutoff location and window, and that its transition matrix has been spectrally analyzed.

The continuity theorem applies conditionally: if the limit profile exists, then the available spectral estimates imply continuity for the relevant range of the profile parameter. The existence of the limit profile itself is left unresolved.