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Large-$N$ Limit of Crossing Probabilities, Discontinuity, and Asymptotic Behavior of Threshold Values in Mandelbrot's Fractal Percolation Process

   
@article{EJP511,
	author = {Erik Broman and Federico Camia},
	title = {Large-$N$ Limit of Crossing Probabilities, Discontinuity, and Asymptotic Behavior of Threshold Values in Mandelbrot's Fractal Percolation Process},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {13},
	year = {2008},
	keywords = {Fractal percolation; crossing probability; critical probability; enhancement/diminishment percolation; near-critical percolation},
	abstract = {
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 $\nu$ for site percolation on the square  lattice, we establish the speed of convergence up to a logarithmic  factor. In particular, our results imply that  $p_c(N,2)-p_c=(\frac{1}{N})^{1/\nu+o(1)}$ as $N \to \infty$, showing  an interesting relation with near-critical percolation.
},
	pages = {no. 33, 980-999},
	issn = {1083-6489},
	doi = {10.1214/EJP.v13-511},    
        url = {http://ejp.ejpecp.org/article/view/511}}