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The invasion percolation process is widely known related to the standard Bernoulli bond percolation model. It is known that the finite cluster density \(\mathbb{P}_{\infty}(p_c)=0\) at critical point for Bernoulli bond percolation in \(\mathbb{Z}^2\), but this conclusion is not yet proven to extend to higher dimensions. We note that it is easy to show the equivalence between \(\mathbb{P}_{\infty}(p_c)=0\) in \(\mathbb{Z}^d\) and \(G_{\mathbb{Z}^d}(0,x)\to0\) as \(\lvert x\rvert\to\infty\) for the invasion percolation process, where \(G_{\mathbb{Z}^d}(0,x)\) is the probability that \(x\) is invaded by an invasion percolation process starting from the origin. In this paper, we will then show by simulation the dominance of \(G_{\mathbb{Z}^3}(0,x)\) by \(G_{\mathbb{Z}^2}(0,x)\) for the same \(\lvert x\rvert\), based on which we show that \(\mathbb{P}_{\infty}(p_c)=0\) in \(\mathbb{Z}^3\). Finally we will numerically estimate the fractal dimension of invasion percolation cluster in \(\mathbb{Z}^3\).
This is my Bachelor of Science thesis for Honors Mathematics and NYU Shanghai. Check out the full paper for more details.