I'm interested in generating a random walk on U(n) using a computer: any references on this topic or related / requisite topics would be helpful. Specific suggestions that discuss the problem technically are welcome as well. Thank you!
Edit: As noted in one of the comments, I'm not looking for trivial random walks. For example, randomly alternating between $U$ and $-U$ would technically be a discrete random walk. I'd like to be able to generate a random walk that can access a dense subset of the group elements; and/or I'd like an analytical description of a continuous random walk (like a diffusion process / Brownian motion), and/or a way to simulate / approximate such a continuous process.