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

Generalizations of Mandelbrot's percolation process are discussed. A process called the random Sierpinski carpet is given particular attention. The process passes through several different phases as its parameters increase from zero to one. The final percolation phase is described in detail. Mandelbrot's implementation of Hoyle's model of galaxies is presented as an example of a percolation. Random sets with a statistically self-similar structure were called canonical curdling by Mandelbrot. Chayes, Chayes and Durrette renamed the process Mandelbrot percolation. Ways of generating the classical Cantor set by means of substitution are shown.