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标题：Fat fractal percolation and k -fractal percolation
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作者：Erik I. Broman ; Tim van de Brug ; Federico Camia 等
期刊名称：Latin American Journal of Probability and Mathematical Statistics
电子版ISSN：1980-0436
出版年度：2012
卷号：IX
期号：2
页码：279
出版社：Instituto Nacional De Matemática Pura E Aplicada
摘要：We consider two variations on theMandelbrot fractal percolationmodel.In the k-fractal percolation model, the d-dimensional unit cube is divided in Ndequal subcubes, k of which are retained while the others are discarded. The procedureis then iterated inside the retained cubes at all smaller scales. We showthat the (properly rescaled) percolation critical value of this model converges tothe critical value of ordinary site percolation on a particular d-dimensional latticeas N ! 1. This is analogous to the result of Falconer and Grimmett (1992) thatthe critical value for Mandelbrot fractal percolation converges to the critical valueof site percolation on the same d-dimensional lattice.In the fat fractal percolation model, subcubes are retained with probability pnat step n of the construction, where (pn)n1 is a non-decreasing Q sequence with 1n=1 pn > 0. The Lebesgue measure of the limit set is positive a.s. given nonextinction.We prove that either the set of connected components larger than onepoint has Lebesgue measure zero a.s. or its complement in the limit set has Lebesguemeasure zero a.s.
关键词：Fractal percolation; random fractals; crossing probability; critical;value.