Cycle structure of riffle shuffles

Abstract

A class of models for riffle shuffles ("$f$-shuffles") related to certain expansive mappings of the unit interval is studied. The main result concerns the cycle structure of the resulting random permutations in $\mathscr{S}_n$ when n is large. It describes the asymptotic distribution of the number of cycles of a given length, relating this distribution to dynamical properties of the associated mapping. This result generalizes a recent result of Diaconis, McGrath and Pitman.