We study the bootstrap and diffusion percolation models in the simple-cubic (sc), body-centered cubic (bcc), and face-centered cubic (fcc) lattices using the Newman-Ziff algorithm. The percolation threshold and critical exponents were calculated through finite-size scaling with high precision in the three lattices. In addition to the continuous and first-order percolation transitions, we found a double transition, which is a continuous transition followed by a discontinuity of the order parameter. We show that the continuous transitions of the bootstrap and diffusion percolation models have the same critical exponents as the classical percolation within error bars and they all belong to the same universality class.

中文翻译：

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三维晶格中的自举和扩散渗透转变

我们使用 Newman-Ziff 算法研究简单立方 (sc)、体心立方 (bcc) 和面心立方 (fcc) 晶格中的自举和扩散渗透模型。渗透阈值和临界指数是通过三个晶格中高精度的有限尺寸缩放计算的。除了连续和一阶渗透转换之外，我们还发现了双重转换，即连续转换后跟阶参数的不连续性。我们表明，自举和扩散渗透模型的连续转换与误差条内的经典渗透具有相同的临界指数，并且它们都属于相同的普遍性类。