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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, theso-called Lyapunov exponent describes the cost of traveling for the simplerandom walk in the potential. The Lyapunov exponent depends on the distributionfunction of the potential, and the aim of this article is to prove that theLyapunov exponent is strictly monotone in the distribution function of thepotential with the order according to strict dominance. Furthermore, thecomparison for the Lyapunov exponent also provides that for the rate functionof this model.

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