Connectivity of the Product Replacement Graph of Simple Groups of Bounded Lie Rank

The Product Replacement Algorithm is a practical algorithm for generating random elements of a finite group. The algorithm can be described as a random walk on a graph whose vertices are the generating k-tuples of the group (for a fixed integer k). We show that there is a function c(r) such that for any finite simple group of Lie type, with Lie rank r, the product replacement graph of the generating k-tuples is connected for any k > c(r). The proof uses results of Larsen and Pink and does not rely on the classification of finite simple groups.