Cutoff for biased transpositions

TL;DR

This paper analyzes the mixing time of a biased transpositions shuffle with two card types, establishing a cutoff at a specific time depending on the bias parameter, using a modified marking scheme.

Contribution

It introduces a new analysis of the biased transpositions shuffle's mixing time and proves a cutoff phenomenon at a precise time depending on the bias parameter.

Findings

Cutoff occurs at time (1/2a) N log N.

Mixing time depends on the bias parameter a.

Modified marking scheme effectively proves the cutoff.

Abstract

In this paper we study the mixing time of a biased transpositions shuffle on a set of $N$ cards with $N/2$ cards of two types. For a parameter $0<a \le 1$, one type of card is chosen to transpose with a bias of $\frac{a}{N}$ and the other type is chosen with probability $\frac{2-a}{N}$. We show that there is cutoff for the mixing time of the chain at time $\frac{1}{2a} N \log N$. Our proof uses a modified marking scheme motivated by Matthews' proof of a strong uniform time for the unbiased shuffle.