Character ratios for exceptional groups of Lie type

We prove character ratio bounds for finite exceptional groups G(q) of Lie type. These take the form |χ(g)|χ(1)≤cqk for all nontrivial irreducible characters χ and nonidentity elements g⁠, where c is an absolute constant, and k is a positive integer. Applications are given to bounding mixing times for random walks on these groups and also diameters of their McKay graphs