Quenched Invariance Principle For Long-Range Reversible Random Walks In Random Environments

In this work, we consider random walks in reversible random environments. The object is to prove quenched invariance principle for the random walks. The contents are divided into two parts.In the first part, we consider a long-range percolation on Z. Under the condition of its percolation exponent s>3, and the nearest neighbors being connected, using the corrector method we prove quenched invariance principle for the simple random walk on the resulted random graph.In the second part, we consider the Bernoulli point process on Zd with parameter p>0which is denoted by V. We construct a random graph with V as vertices set, and all nearest neighbors in V along each coordinate direction as edges. Let each edge be associated with positive random variable which is referred as conductance. Suppose all the conductances are i.i.d. and independent with the Bernoulli process. Assuming that the conductances are bounded away from0, we consider random walks in these random environments under two cases according to the conductances are bounded from above or not.Firstly, under bounded conductances, using the corrector method we prove quenched invariance principle. To this end, we also prove almost sure gaussian upper bounds for the transition probabilities of random walks.For unbounded conductances, under the extra assumptions that the dimen-sion d>1, using the corrector method we prove quenched invariance principle. To this end, we prove almost sure gaussian upper bounds and lower bounds for the transition probabilities of random walks. We also give estimates for a first-passage percolation on the constructed random graph.Using theory of electrical networks, we give another construction of correc-tor.