I will talk about large deviations for nearest-neighbor random walk in an i.i.d. environment on $Z^d$. There exist variational formulae for the quenched and the averaged rate functions $I_q$ and $I_a$, obtained by Rosenbluth and Varadhan, respectively. $I_q$ and $I_a$ are not identically equal. However, when $d\geq4$ and the walk satisfies the so-called (T) condition of Sznitman,...