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 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 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) 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 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.Comment: 27 pages, 3 figure