Strict comparison for the Lyapunov exponents of the simple random walk in random potentials

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. In particular, for the one-dimensional annealed situation, we observe that the Lyapunov exponents can coincide even under the strict dominance. Furthermore, the comparison for the Lyapunov exponent also provides that for the rate function of this model.