New methods to bound the critical probability in fractal percolation

Henk Don

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arXiv · 1210.4150

New methods to bound the critical probability in fractal percolation

Henk Don

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Abstract

Fractal percolation has been introduced by Mandelbrot in 1974. We study the two-dimensional case, with integer subdivision index M and survival probability p. It is well known that there exists a non-trivial critical value p_c(M) such that a.s. the largest connected component in the limiting set K is a point for p 0.881 and p_c(3)>0.784. For the upper bounds, we introduce the idea of classifications. The fractal percolation iteration process now induces an iterative random process on a finite alphabet, which is easier to analyze than the original process. This theoretical framework is the basis of computer aided proofs for the following upper bounds: p_c(2)<0.993, p_c(3)<0.940 and p_c(4)<0.972.

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Henk Don. 2012-10-15. New methods to bound the critical probability in fractal percolation. https://arxiv.org/abs/1210.4150

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