Fat fractal percolation and k-fractal percolation
2012 (English)In: Latin American Journal of Probability and Mathematical Statistics, ISSN 1980-0436, Vol. 9, no 2, 279-301 p.Article in journal (Refereed) Published
We consider two variations on the Mandelbrot fractal percolation model. In the k-fractal percolation model, the d-dimensional unit cube is divided in N-d equal subcubes, k of which are retained while the others are discarded. The procedure is then iterated inside the retained cubes at all smaller scales. We show that the (properly rescaled) percolation critical value of this model converges to the critical value of ordinary site percolation on a particular d-dimensional lattice as N -> infinity. This is analogous to the result of Falconer and Grimmett (1992) that the critical value for Mandelbrot fractal percolation converges to the critical value of site percolation on the same d-dimensional lattice. In the fat fractal percolation model, subcubes are retained with probability p(n) at step n of the construction, where (p(n))(n >= 1) is a non-decreasing sequence with Pi(infinity)(n=1) p(n) > 0. The Lebesgue measure of the limit set is positive a.s. given non-extinction. We prove that either the set of connected components larger than one point has Lebesgue measure zero a.s. or its complement in the limit set has Lebesgue measure zero a.s.
Place, publisher, year, edition, pages
2012. Vol. 9, no 2, 279-301 p.
Fractal percolation, random fractals, crossing probability, critical value
IdentifiersURN: urn:nbn:se:uu:diva-208182ISI: 000323683400002OAI: oai:DiVA.org:uu-208182DiVA: diva2:651169