Consider the matrix exponential map $H \mapsto e^{i H t}$ acting on the Gaussian unitary ensemble (GUE) of Hermitian matrices.
I would expect that for large $t$, the resulting measure on the unitary group approaches Haar measure -- is that right? Is there a simple heuristic argument showing that the limiting distribution is unitary-translation-invariant?
I imagine that some much larger class of distributions on the Lie algebra might also yield the Haar measure under exponentiation for large $t$? (Perhaps for other Lie groups / Lie algebras as well?) And if so, I'm also curious which distribution converges most quickly to Haar measure (for some fixed normalization of the size of $H$).