  We consider the simple random walk in i.i.d. nonnegative potentials on the
$d$-dimensional cubic lattice $\mathbb{Z}^d$ ($d \geq 1$). In this model, the
so-called Lyapunov exponent describes the cost of traveling for the simple
random walk in the potential. The Lyapunov exponent depends on the distribution
function of the potential, and the aim of this article is to prove that the
Lyapunov exponent is strictly monotone in the distribution function of the
potential with the order according to strict dominance. Furthermore, the
comparison for the Lyapunov exponent also provides that for the rate function
of this model.
