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- W2077971405 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<p_c(M) and with positive probability there is a connected component intersecting opposite sides of the unit square for pgeq p_c(M). For all Mgeq 2, the value of p_c(M) is unknown. In this paper we present ideas to find lower and upper bounds, significantly sharper than those already known. To find lower bounds, we compare fractal percolation with site percolation. A fundamentally new result is that for all M we construct an increasing sequence that converges to p_c(M). The terms in the sequence can in principle be calculated algorithmically. These ideas lead to (computer aided) proofs that p_c(2)> 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." @default.
- W2077971405 created "2016-06-24" @default.
- W2077971405 creator A5074680417 @default.
- W2077971405 date "2014-07-31" @default.
- W2077971405 modified "2023-09-26" @default.
- W2077971405 title "New methods to bound the critical probability in fractal percolation" @default.
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- W2077971405 doi "https://doi.org/10.1002/rsa.20566" @default.
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