Shuffling via Transpositions
Samira Arfaee, Evita Nestoridi
TL;DR
The paper analyzes a family of card shuffles generated by transpositions corresponding to Jucys–Murphy elements, providing a complete spectral characterization via the lifting eigenvectors framework. It establishes a sharp total-variation cutoff for the $k$-star transpositions at time $t_{n,k}(c)=\frac{2n-(k+1)}{2(n-1)}\,n\(\log n+c\)$ with window $\frac{2n-(k+1)}{2(n-1)}\,n$, and derives precise upper and lower bounds that confirm the cutoff. The limit profile is shown to coincide with the random-transpositions profile when $k/n\to 0$ or $1$, extending the understanding of mixing for transposition-based shuffles through representation-theoretic methods. The work unifies diagonalization via lifting operators with detailed eigenvalue bounds to deliver a rigorous spectral analysis of $k$-star transpositions.
Abstract
We consider a family of card shuffles of $n$ cards, where the allowed moves involve transpositions corresponding to the Jucys--Murphy elements of $\{S_m\}_{m \leq n}$. We diagonalize the transition matrix of these shuffles. As a special case, we consider the $k$-star transpositions shuffle, a natural interpolation between random transpositions and star transpositions. We proved that the $k$-star transpositions shuffle exhibits total variation cutoff at time $\frac{2n-(k+1)}{2(n-1)}n\log n$ with a window of $\frac{2n-(k+1)}{2(n-1)}n$. Furthermore, we prove that for the case where $k/n \rightarrow 0$ or $1$, this shuffle has the same limit profile as random transpositions, which has been fully determined by Teyssier.