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Keane, Michael






Take a porous stone and immerse it in water. Does the center of the stone contain water? The answer to this question can be given by understanding mathematical percolation. The percolation process is a model of an event passing through a space. The event can be one such as a disease spreading, or such as an object being permeated by a liquid. The space restricts this. In the pure mathematical sense of the percolation process, we care not about the event, but its ability to span. To be able to handle this problem, we consider our space to be in the form of a random lattice. We mostly work with the square lattice, but will mention some others such as the triangular lattice. There are two types of mechanics used to describe how our liquid penetrates through our lattice. In site percolation, each vertex has a probability (independent of the rest) of being 'open', otherwise it is 'closed'. Our liquid is allowed to pass to an adjacent vertex only if it is open. Bond percolation considers edges in our lattice open or closed and events travel from one vertex to an adjacent vertex only if the edge is open. At some probability level, our liquid will succeed in reaching an infinite number of vertices. This leads us to the idea of a critical probability (or $p_c$), where if our vertices/edges have a probability greater than $p_c$ of being open, then with positive probability the liquid will reach an infinite number of vertices. On the other hand, if the probability that a vertex or edge is open is less than our critical probability, then surely the event will not be able to reach infinitely many vertices. In chapter 2 we consider percolation on a finite rectangular sub-lattice. After determining which edges or vertices are closed in our finite lattice, we 'copy' the outcome and periodically layer the entire plane with these copies. We call this determinate percolation as one sub-lattice or 'tile' determines the outcome of the entire lattice. This is a new model which I have not been able to find in the literature, and which exhibits apparently some interesting phenomena.



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