Top to random shuffles on colored permutations

TL;DR

This paper analyzes a Markov chain on signed and colored permutations generated by a top-to-random shuffle with flipping, determining eigenvalues, multiplicities, and showing the mixing time is on the order of n log n.

Contribution

It extends the analysis of top-to-random shuffles to signed and colored permutations, deriving eigenvalues, multiplicities, and mixing times for these generalized processes.

Findings

Eigenvalues are 0, i/n for i=0,...,n with specific multiplicities.

Mixing time is approximately n log n, matching the classical case.

Cut-off phenomenon is characterized using Stirling numbers asymptotics.

Abstract

A deck of $n$ cards are shuffled by repeatedly taking off the top card, flipping it with probability $1/2$, and inserting it back into the deck at a random position. This process can be considered as a Markov chain on the group $B_n$ of signed permutations. We show that the eigenvalues of the transition probability matrix are $0,1/n,2/n,\ldots,(n-1)/n,1$ and the multiplicity of the eigenvalue $i/n$ is equal to the number of the {\em signed} permutation having exactly $i$ fixed points. We show the similar results also for the colored permutations. Further, we show that the mixing time of this Markov chain is $n\log n$, same as the ordinary 'top-to-random' shuffles without flipping the cards. The cut-off is also analyzed by using the asymptotic behavior of the Stirling numbers of the second kind.