We study the connectivity properties of the complementary set in Poisson multiscale percolation model and in Mandelbrot's percolation model in arbitrary dimension. By using a result about majorizing dependent random fields by Bernoulli fields, we prove that if the selection parameter is less than certain critical value, then, by choosing the scaling parameter large enough, we can assure that there is no percolation in the complementary set.
Menshikov, M., Yu, P. S., & Vachkovskaia, M. (2001). On the connectivity properties of the complementary set in fractal percolation models. Probability Theory and Related Fields, 119(2), 176-186. https://doi.org/10.1007/pl00008757