Mandelbrot Percolation Research Articles

On the structure of Mandelbrot's percolation process and other random cantor sets

We consider generalizations of Mandelbrot's percolation process. For the process which we call the random Sierpinski carpet, we show that it passes through several different phases as its parameter increases from zero to one. The final section treats the percolation phase.

Connectivity properties of Mandelbrot's percolation process

In 1974, Mandelbrot introduced a process in [0, 1]2 which he called “canonical curdling” and later used in this book(s) on fractals to generate self-similar random sets with Hausdorff dimension D∈(0,2). In this paper we will study the connectivity or “percolation” properties of these sets, proving all of the claims he made in Sect. 23 of the “Fractal Geometry of Nature” and a new one that he did not anticipate: There is a probability pc∈(0,1) so that if p<pc then the set is “duslike” i.e., the largest connected component is a point, whereas if p≧pc (notice the =) opposing sides are connected with positive probability and furthermore if we tile the plane with independent copies of the system then there is with probability one a unique unbounded connected component which intersects a positive fraction of the tiles. More succinctly put the system has a first order phase transition.