The random \(k\) cycle walk on the symmetric group

We study the random walk on the symmetric group \(S_n\) generated by the conjugacy class of cycles of length \(k\). We show that the convergence to uniform measure of this walk has a cut-off in total variation distance after \(\frac{n}{k} log n\) steps, uniformly in \(k = o(n)\) as \(n \to \infty\). The analysis follows from a new asymptotic estimation of the characters of the symmetric group evaluated at cycles.