  We consider large deviations for nearest-neighbor random walk in a uniformly
elliptic i.i.d. environment on $\mathbb{Z}^d$. There exist variational formulae
for the quenched and 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, they have been previously shown to be equal on an open
set $\mathcal{A}_{\mathit {eq}}$. For every $\xi\in\mathcal{A}_{\mathit {eq}}$,
we prove the existence of a positive solution to a Laplace-like equation
involving $\xi$ and the original transition kernel of the walk. We then use
this solution to define a new transition kernel via the h-transform technique
of Doob. This new kernel corresponds to the unique minimizer of Varadhan's
variational formula at $\xi$. It also corresponds to the unique minimizer of
Rosenbluth's variational formula, provided that the latter is slightly
modified.
