Babai’s conjecture for high-rank classical groups with random generators

Let $$G = {\text {SCl}}_n(q)$$ G = SCl n ( q ) be a quasisimple classical group with n large, and let $$x_1, \ldots , x_k \in G$$ x 1 , … , x k ∈ G be random, where $$k \ge q^C$$ k ≥ q C . We show that the diameter of the resulting Cayley graph is bounded by $$q^2 n^{O(1)}$$ q 2 n O ( 1 ) with probability $$1 - o(1)$$ 1 - o ( 1 ) . In the particular case $$G = {\text {SL}}_n(p)$$ G = SL n ( p ) with p a prime of bounded size, we show that the same holds for $$k = 3$$ k = 3 .