#### 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)=(nP_{n+1}+(n+1)P_{n})/(4nQ_{n}) where P_{n} and Q_{n} are the n-th Pell and Pell-Lucas numbers respectively. Moreover we show that as n→∞, ρ(∞)=(1 + sqrt(2))/8.

#### Recommended Citation

Rosell Gonzalez, Pablo, "Limiting Behavior of Nondeterministic Fillings of the Torus by Colored Squares" (2011). *Theses and Dissertations*. 124.

https://scholarworks.uark.edu/etd/124