The random k cycle walk on the symmetric group

الأصل · EN

We study the random walk on the symmetric group Sₙ 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 n/k log n steps, uniformly in k = o(n) as n → ∞. The analysis follows from a new asymptotic estimation of the characters of the symmetric group evaluated at cycles.