  We consider the simple random walk on supercritical percolation clusters in
the multidimensional cubic lattice. In this model, a quenched large deviation
principle holds for the position of the random walk. Its rate function depends
on the law of the percolation configuration, and the aim of this paper is to
study the continuity of the rate function in the law. To do this, it is useful
that the rate function is expressed by the so-called Lyapunov exponent, which
is the asymptotic cost paid by the random walk for traveling in a landscape of
percolation configurations. In this context, we first observe the continuity of
the Lyapunov exponent in the law of the percolation configuration, and then
lift it to the rate function.
