The mixing time and spectral gap of a random walk on the symmetric group can sometimes be understood in terms of its low dimensional representations (e.g., Aldous' spectral gap conjecture). It turns out that under a mild degree condition involving the step of the group, the same holds for nilpotent groups w.r.t. their one dimensional representations: the spectral gap and the epsilon total variation mixing time of the walk on G are determined by those of the projection of the walk to the abelianization G/[G,G]. We'll discuss some applications concerning the cutoff phenomenon (= abrupt convergence to equilibrium) and the dependence (or lack of!) of the spectral gap and the mixing time on the choice of generators.
As time permits we shall discuss a related result, confirming in the nilpotent setup a conjecture of Aldous and Diaconis concerning the occurrence of cutoff when a diverging number of generators are picked uniformly at random. Joint work with Zoe Huang.
