  We consider the simple random walk on the $d$-dimensional lattice
$\mathbb{Z}^d$ ($d \geq 1$), traveling in potentials which are Bernoulli
distributed. The so-called Lyapunov exponent describes the cost of traveling
for the simple random walk in the potential, and it is known that the Lyapunov
exponent is strictly monotone in the parameter of the Bernoulli distribution.
Hence, the aim of this paper is to investigate the effect of the potential on
the Lyapunov exponent more precisely, and we derive some Lipschitz-type
estimates for the difference between the Lyapunov exponents.
