Scaling limits for one-dimensional long-range percolation: Using the corrector method

In this paper, by using the corrector method we give another proof of the quenched invariance principle for the random walk on the infinite random graph generated by a one-dimensional long-range percolation under the conditions that the connection probability p(1)=1p(1)=1 and the percolation exponent s>2s>2. The key step of the proof is the construction of the corrector. We show that the corrector can be constructed under either s∈(2,3]s∈(2,3] or s>3s>3, though the corresponding underlying measures may be different. As an application of the main result we get a new lower bound of the quenched diagonal transition probability for the random walk.