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Large-$$N$$ limit of crossing probabilities, discontinuity, and asymptotic behavior of threshold values in Mandelbrot’s fractal percolation process. (English) Zbl 1191.60109
Summary: 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 $$p_c(N,2)-p_c=(1/N)^{1/\nu +o(1)}$$ as $$N\to \infty$$, showing an interesting relation with near-critical percolation.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B43 Percolation
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