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

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Sarah J. Suddendorf

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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.