Cutoff for biased transpositions

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 $0u003ca le 1$, 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 Matthewsu0027 proof of a strong uniform time for the unbiased shuffle.