Cutoff for the Biased Random Transposition Shuffle

TL;DR

This paper analyzes the biased random transposition shuffle, proving it exhibits a sharp cutoff at a specific mixing time and characterizing the distribution of fixed cards near that cutoff.

Contribution

It provides the first diagonalization of the transition matrix for the biased shuffle and establishes the cutoff time and distributional limits.

Findings

Total variation cutoff at time t_N = (1/2b) N log N

Eigenvalues used to determine mixing time

Limiting distribution of fixed cards is Poisson

Abstract

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.