Mandelbrot分形逾渗过程中交叉概率的大N极限、不连续性和门限值的渐近性态
Large-N Limit of Crossing Probabilities, Discontinuity, and Asymptotic Behavior of Threshold Values in Mandelbrot's Fractal Percolation Process
摘要：
研究了$d \geq 2$维上的Mandelbrot渗流过程.该过程通过迭代过程产生随机分形集，该过程开始于将单位立方体$[0,1]^d$划分为$N^d$子立方体，并分别以概率$p$或$1-p$独立地保留或丢弃每个子立方体。然后在所有尺度下在保留的子立方体内重复该步骤。当$p$变化时，就所有$d \geq 2$的路径而言，以及就所有$d \geq 3$的$(d-1)$ -维“片”而言，存在渗透相变。对于任何$d \geq 2$，我们认为随机分形集产生的路径渗流临界值$p_c(N,d)$，并表明，它包含一个路径连接两个相对的面的立方体$[0,1]^d$的概率趋于一作为$N \to \infty$。作为直接的结果，我们得到，上述概率具有不连续性，作为$p$的函数，在$p_c(N,d)$为所有$N$足够大。这在以前只对$d=2$（对任何$N \geq 2$）证明过。对于$d \geq 3$，我们证明了类似的结果，片渗流。在二维中，Chayes和Chayes证明了$p_c(N,2)$收敛于正方形晶格上的位点渗透的临界密度$p_c$，即$N \to \infty$。假设存在的相关长度指数$ν$的网站渗流的正方形格子，我们建立的收敛速度的对数因子。特别是，我们的研究结果意味着，$p_c(N,2)-p_c=(\frac{1}{N})^{1/ν+o(1)}$作为$N \to \infty$，显示出一个有趣的关系与近临界渗流。
英文摘要：
We study Mandelbrot's percolation process in dimension $d \geq 2$. The process generates random fractal sets by an iterative procedure which starts by dividing the unit cube $[0,1]^d$ in $N^d$ subcubes, and independently retaining or discarding each subcube with probability $p$ or $1-p$ respectively. This step is then repeated within the retained subcubes at all scales. As $p$ is varied, there is a percolation phase transition in terms of paths for all $d \geq 2$, and in terms of $(d-1)$-dimensional "sheets" for all $d \geq 3$. For any $d \geq 2$, we consider the random fractal set produced at the path-percolation critical value $p_c(N,d)$, and show that the probability that it contains a path connecting two opposite faces of the cube $[0,1]^d$ tends to one as $N \to \infty$. As an immediate consequence, we obtain that the above probability has a discontinuity, as a function of $p$, at $p_c(N,d)$ for all $N$ sufficiently large. This had previously been proved only for $d=2$ (for any $N \geq 2$). For $d \geq 3$, we prove analogous results for sheet-percolation. In dimension two, Chayes and Chayes proved that $p_c(N,2)$ converges, as $N \to \infty$, to the critical density $p_c$ of site percolation on the square lattice. Assuming the existence of the correlation length exponent $ν$ for site percolation on the square lattice, we establish the speed of convergence up to a logarithmic factor. In particular, our results imply that $p_c(N,2)-p_c=(\frac{1}{N})^{1/ν+o(1)}$ as $N \to \infty$, showing an interesting relation with near-critical percolation.
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