These are examples of multiplicative cascade fractals. To create these distributions there are a few steps to take. Firstly, we must create a lattice of points which will be our underlying probability density field. Then we will populate this lattice with randomly placed points, insisting that the probability that the points be placed are proportional to the cell probability.

The fractal is constructed as follows: The space is split into four equal parts, each part is then assigned a probability from the set ${\displaystyle \lbrace p_{1},p_{2},p_{3},p_{4}\rbrace }$ without replacement. Where ${\displaystyle p_{i}\in [0,1]}$. Each subspace is then divided again and assigned probabilities randomly from the same set and this is continued to the ${\displaystyle N^{th}}$ level.

At the ${\displaystyle N^{th}}$ level the probability of a cell being occupied is the product of the cell's ${\displaystyle p_{i}}$ and its parents and ancestors up to level 1 i.e. all the cells above it. In constructing this model down to level 8 we produce a ${\displaystyle 4^{8}}$ array of cells each with its own probability. To then place particle in the space we invoke a Monte Carlo rejection scheme. Choosing x and y coordinates randomly we simply test if a random number between 0 and 1 is less or greater than the cell probability. We typically dust the probability density field with 5,000 points in a space of 256 x 256.

Model I (left plot):${\displaystyle \lbrace p_{1},p_{2},p_{3},p_{4}\rbrace =\lbrace 1,1,1,0\rbrace }$ Model II (middle plot):${\displaystyle \lbrace p_{1},p_{2},p_{3},p_{4}\rbrace =\lbrace 1,0.75,0.75,0.5\rbrace }$

${\displaystyle \lbrace p_{1},p_{2},p_{3},p_{4}\rbrace =\lbrace 1,0.5,0.5,0.25\rbrace }$