Date of Graduation

8-2011

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level

Graduate

Department

Mathematical Sciences

Advisor

Chaim Goodman-Strauss

Committee Member

Mark E. Arnold

Second Committee Member

Russell J. Deaton

Third Committee Member

Yo'av Rieck

Keywords

Applied sciences, Pure sciences, Colored squares, Discrete mathematics, Limiting behavior, Nondeterministic fillings, Torus

Abstract

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 + sqrt(2))/8.

Share

COinS