Cyclic to Random Transposition Shuffles

TL;DR

This paper analyzes two variants of cyclic to random transposition shuffles, deriving explicit limiting density functions for the distribution of card positions and numbers, revealing discontinuities and extremal behaviors.

Contribution

It provides explicit formulas for the limiting distributions of card positions and labels in two new cyclic to random transposition shuffles, including analysis of their discontinuities.

Findings

Derived explicit limiting density functions for the shuffles

Identified discontinuities at the point x=b in the density functions

Showed extremal density values are approached near the discontinuities

Abstract

Consider a permutation $\sigma\in S_n$ as a deck of cards numbered from 1 to $n$ and laid out in a row, where $\sigma_j$ denotes the number of the card that is in the $j$-th position from the left.\rm\ We define two cyclic to random transposition shuffles. The first one works as follows: for $j=1,..., n$, on the $j$-th step transpose the card that was \it originally\rm\ the $j$-th from the left with a random card (possibly itself). The second shuffle works as follows: on the $j$-th step, transpose the card that is \it currently\rm\ in the $j$-th position from the left with a random card (possibly itself). For these shuffles, for each $b\in[0,1]$, we calculate explicitly the limiting rescaled density function of $x,0\le x\le1$, for the probability that a card with a number around $bn$ ends up in a position around $xn$, and for each $x\in[0,1]$, we calculate the limiting rescaled density function of $b,0\le b\le 1$, for the probability that the card in a position around $xn$ will be a card with a number around $bn$. These density functions all have a discontinuity at $x=b$, and for each of them, the supremum of the density is obtained by approaching the discontinuity from one side, and, for certain values of the parameter, the infimum of the density is obtained by approaching the discontinuity from the other side.