Behavior of random walk on discrete point processes

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We consider a model for random walks on random environments (RWRE) with random subset of Zᵈ as the vertices, and uniform transition probabilities on 2d points (two "coordinate nearest points" in each of the d coordinate directions). We give partial characterization of transience and recurrence in the different dimensions. Finally we prove Central Limit Theorem (CLT) for such random walks, under a condition on the distance between coordinate nearest points.