Random Lie bracket on $\mathfrak{sl}_2(\mathbf{F}_p)$

We study a random walk on the Lie algebra $\mathfrak{sl}_2(\mathbf{F}_p)$ where new elements are produced by randomly applying adjoint operators of two generators. Focusing on the generic case where the generators are selected at random, we analyze the limiting distribution of the random walk and the speed at which it converges to this distribution. These questions reduce to the study of a random walk on a cyclic group. We show that, with high probability, the walk exhibits a pre-cutoff phenomenon after roughly $p$ steps. Notably, the limiting distribution need not be uniform and it depends on the prime divisors of $p-1$. Furthermore, we prove that by incorporating a simple random twist into the walk, we can embed a well-known affine random walk on $\mathbf{F}_p$ into the modified random Lie bracket, allowing us to show that the entire Lie algebra is covered in roughly $\log p$ steps in the generic case.