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Limiting Behavior of Nondeterministic Fillings of the Torus by Colored Squares

In this work we study different dynamic processes for filling tori and n ×∞ bands with edge-to-edge black and white squares at random. First we present a simulation for the Random Sequential Adsorption (RSA) with nearest-neighbor rejection on n ×n tori. We are interested in the ratio of black to total tiles once the domain is saturated for large domains. Next we study the annealing process. Given a random excited tiling of an n ×n torus, we show that as t →∞ the system reaches a stable state in which no tile is excited. This stable state can either be a tiling whose tiles are all the same color, or is formed by vertical or horizontal strips of alternating colors. The third process consists of stamping a d-dimensional nd torus with a stamp consisting of a finite number S of colored d-cubes replacing, at each time t , the tiles of the domain by the stamp at a position chosen randomly with uniform distribution. We show that the ratio of a particular color of cubes c to nd remains “close to” the ratio of c cubes on the stamp to S. Finally we analyze the accretion of layers of width n that satisfy nearest-neighbor rejection and guarantee saturation. In this case we show that the expected ratio of black tiles to the total number of tiles, as time t →∞ is given by ρ(n) = (nPn+1 + (n + 1)Pn)/(4nQn) where Pn and Qn are the n-th Pell and Pell-Lucas numbers respectively. Moreover we show that as n →∞, ρ(∞) = (1+ p 2)/8. This dissertation is approved for recommendation to the Graduate Council Dissertation Director: Dr. Chaim Goodman-Strauss Dissertation Committee: