Fat fractal percolation and k-fractal percolation

TL;DR

This paper investigates two variations of fractal percolation models, analyzing their critical values and measure properties, and establishes convergence results and measure dichotomies for the limit sets.

Contribution

It introduces and analyzes k-fractal and fat fractal percolation models, showing convergence of critical values and measure dichotomies, extending previous results on Mandelbrot fractal percolation.

Findings

Critical percolation value converges to the site percolation threshold as N increases.

In fat fractal percolation, the limit set's measure is positive with non-zero probability.

The set of large connected components or its complement has Lebesgue measure zero almost surely.

Abstract

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.