Let $B_t$ be the unitary Brownian motion, i.e. Brownian motion on the unitary group $U(N)$. What is known about the mixing time of $B_t$, that is, how fast does the measure $B_t(\delta_{\{Id\}})$ converge to the Haar measure? (the question is motivated by trying to say something about mixing times for some discrete approximations of unitary Brownian motion)
If I understand correctly after a cursory check of the literature, the mixing time for Brownian motion on a compact simple Lie group $G$ is conjectured to behave asymptotically (in the limit of large dimension) as $\log \dim G$. See section 3 (and 3.2 in particular) of this paper. 

