Character bounds for finite groups of Lie type

We establish new bounds on character values and character ratios for finite groups $G$ of Lie type, which are considerably stronger than previously known bounds, and which are best possible in many cases. These bounds have the form $\mathrm{\mid }\chi (g)\mathrm{\mid }\le \chi (1{)}^{{\alpha }_g}$, and give rise to a variety of applications, for example to covering numbers and mixing times of random walks on such groups. In particular we deduce that, if $G$ is a classical group in dimension $n$, then, under some conditions on $G$ and $g\in G$, the mixing time of the random walk on $G$ with the conjugacy class of $g$ as a generating set is (up to a small multiplicative constant) $n\mathrm{/}s$, where $s$ is the support of $g$.