Riffle shuffles with biased cuts

TL;DR

This paper studies a biased cut variant of the riffle shuffle model, extending previous work, and demonstrates a sharp cutoff in mixing times using connections to quasisymmetric functions and complex analysis.

Contribution

It introduces a biased cut model for riffle shuffles and provides a rigorous analysis of its mixing behavior, extending prior results to this new setting.

Findings

Sharp cutoff in separation distance

Sharp cutoff in L-infinity distance

Analysis based on quasisymmetric functions and complex analysis

Abstract

The well-known Gilbert-Shannon-Reeds model for riffle shuffles assumes that the cards are initially cut 'about in half' and then riffled together. We analyze a natural variant where the initial cut is biased. Extending results of Fulman (1998), we show a sharp cutoff in separation and L-infinity distances. This analysis is possible due to the close connection between shuffling and quasisymmetric functions along with some complex analysis of a generating function.