Good to the Last Drop: Percolation Through Hypercubic and Random Lattices for Integer, Fractional, and Fractal Dimensions

Sarah J. Suddendorf

2007

Percolation is the gradual movement of a liquid through a porous material, and is of vast interest to both mathematicians and physicist. This thesis investigates percolation in different dimensions through several computer simulations that were written to collect the data. The difference between a top flooded grid and a center flooded grid was investigated and was found to be negligible. The slope, or step size of the critical phenomena was studied and found to have a dependence relation with the grid size. The critical probabilities were found for integer dimensions from two through five. Fractal and fractional dimensions were also investigated, and the critical probabilities were found experimentally. Lastly, percolation was studied for a random grid. On the random grid, the percolation probability did not only depend on the vacancy probability, but rather it depended on two variables. Instead of finding a single critical probability, there was a critical relationship between the two variables. The data collected supports the theories predicted for the areas studied of this thesis.