The Random $$(n-k)$$ ( n - k ) -Cycle to Transpositions Walk on the Symmetric Group

Abstract

We study the rate of convergence of the Markov chain on $$S_n$$ which starts with a random $$(n-k)$$ -cycle for a fixed $$k \ge 1$$ , followed by random transpositions. The convergence to the stationary distribution turns out to be of order n. We show that after $$cn + \frac{\ln k}{2}n$$ steps for $$c>0$$ , the law of the Markov chain is close to the uniform distribution. The character of the defining representation is used as test function to obtain a lower bound for the total variation distance. We identify the asymptotic distribution of the test function given the law of the Markov chain for the $$(n-1)$$ -cycle case. The upper bound relies on estimates for the difference of normalized characters.