Random walks generated by the Ewens distribution on the symmetric group

This paper studies Markov chains on the symmetric group $S_n$ where the transition probabilities are given by the Ewens distribution with parameter $\theta>1$. The eigenvalues are identified to be proportional to the content polynomials of partitions. We show that the mixing time is bounded above by a constant depending only on the parameter if $\theta$ is fixed. However, if it agrees with the number of permuted elements ($\theta=n$), the sequence of chains has a total variation cutoff at $\frac{\log n}{\log 2}.$