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In this article, we focus on Bernoulli percolation and mainly investigate bounds of the probability of the connectivity of 0 to the distance n. At first, we give a rough bound of the probability, and then refine our result by the high-dimensional RSW theory, which gives a nontrivial bound for a short crossing in ℤ𝒅, as well as the renormalization method. We finish the last step of this section by coupling. Next, we give a more refined bound in ℤ𝟐 using the dual graph. At last, we investigate the behaviors of some subgraphs of ℤ𝟐. The work done and conclusions made in this article are fundamental in percolation theory. The purpose of this article is to introduce Bernoulli percolation in a relatively thorough way to the beginners of percolation theory and provide some insights into different proofs of basic conclusions.
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