Probabilistic Waring problems for finite simple groups

The probabilistic Waring problem for finite simple groups asks whether every word of the form $w_1w_2$, where $w_1$ and $w_2$ are non-trivial words in disjoint sets of variables, induces almost uniform distribution on finite simple groups with respect to the $L^1$ norm. Our first main result provides a positive solution to this problem. We also provide a geometric characterization of words inducing almost uniform distribution on finite simple groups of Lie type of bounded rank, and study related random walks. Our second main result concerns the probabilistic $L^{\infty}$ Waring problem for finite simple groups. We show that for every $l \ge 1$ there exists $N = N(l)$, such that if $w_1, \ldots , w_N$ are non-trivial words of length at most $l$ in pairwise disjoint sets of variables, then their product $w_1 \cdots w_N$ is almost uniform on finite simple groups with respect to the $L^{\infty}$ norm. The dependence of $N$ on $l$ is genuine. This result implies that, for every word $w = w_1 \cdots w_N$ as above, the word map induced by $w$ on a semisimple algebraic group over an arbitrary field is a flat morphism. Applications to representation varieties, subgroup growth, and random generation are also presented.