Cutoff for the Biased Random Transposition Shuffle

In this paper, we study the biased random transposition shuffle, a natural generalization of the classical random transposition shuffle studied by Diaconis and Shahshahani. We diagonalize the transition matrix of the shuffle and use these eigenvalues to prove that the shuffle exhibits total variation cutoff at time $t_N=\frac{1}{2b}N\log N$ with window $N$. We also prove that the limiting distribution of the number of fixed cards near the cutoff time is Poisson.