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We consider the simple random walk on supercritical percolation clusters inthe multidimensional cubic lattice. In this model, a quenched large deviationprinciple holds for the position of the random walk. Its rate function dependson the law of the percolation configuration, and the aim of this paper is tostudy the continuity of the rate function in the law. To do this, it is usefulthat the rate function is expressed by the so-called Lyapunov exponent, whichis the asymptotic cost paid by the random walk for traveling in a landscape ofpercolation configurations. In this context, we first observe the continuity ofthe Lyapunov exponent in the law of the percolation configuration, and thenlift it to the rate function.

Continuity result for the rate function of the simple random walk on supercritical percolation clusters