Fat fractal percolation and k-fractal percolation

Details Fat fractal percolation and k-fractal percolation Fat fractal percolation and k-fractal percolation

2011-11-29

arxiv.org/abs/1111.6834v1

Mathematics

Probability

26 pages, 3 figures

Scientific paper

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 site percolation in L^d as N tends to infinity. This is analogous to the result of Falconer and Grimmett that the critical value for Mandelbrot fractal percolation converges to the critical value of site percolation in L^d. In the fat fractal percolation model, subcubes are retained with probability p_n at step n of the construction, where (p_n) is a non-decreasing sequence with \prod p_n > 0. The Lebesgue measure of the limit set is positive a.s. given non-extinction. We show that with probability 1 either the set of "dust" points or the set of connected components larger than one point has positive Lebesgue measure, but never both.

Affiliated with

Also associated with

No associations

Rating

Fat fractal percolation and k-fractal percolation does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Fat fractal percolation and k-fractal percolation, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Fat fractal percolation and k-fractal percolation will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-16557