# Mathematics Colloquia and Seminars

Consider large deviations for nearest-neighbor random walk in a uniformly elliptic i.i.d. environment. Denote the quenched and averaged rate functions by $I_q$ and $I_a$, respectively. Note that $I_a$ is less than or equal to $I_q$ by Jensen's inequality. Also, it is easy to see that $I_q$ and $I_a$ are not identically equal. In this talk, I will present two results: (1) For ballistic walks in dimensions four or more, $I_q$ and $I_a$ are equal on a closed set whose interior contains every nonzero velocity at which the rate functions vanish. (2) The first result is not valid in general for ballistic walks in lower dimensions. (Joint work with O. Zeitouni.)