The overhand shuffle mixes in Θ(n2logn) steps

Abstract

The overhand shuffle is one of the “real” card shuffling methods in the sense that some people actually use it to mix a deck of cards. A mathematical model was constructed and analyzed by Pemantle [J. Theoret. Probab. 2 (1989) 37–49] who showed that the mixing time with respect to variation distance is at least of order n² and at most of order n²logn. In this paper we use an extension of a lemma of Wilson [Ann. Appl. Probab. 14 (2004) 274–325] to establish a lower bound of order n²logn, thereby showing that n²logn is indeed the correct order of the mixing time. It is our hope that the extension of Wilson’s lemma will prove useful also in other situations; it is demonstrated how it may be used to give a simplified proof of the Θ(n³logn) lower bound of Wilson [Electron. Comm. Probab. 8 (2003) 77–85] for the Rudvalis shuffle.