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.
","abstract_html":"<p>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<sub>n+1</sub>+(n+1)P<sub>n</sub>)/(4nQ<sub>n</sub>) where P<sub>n</sub> and Q<sub>n</sub> are the n-th Pell and Pell-Lucas numbers respectively. Moreover we show that as n→∞, ρ(∞)=(1 + sqrt(2))/8.</p>","abstract_has_math":false,"creators":["Rosell Gonzalez, Pablo"],"institution":null,"degree_name":"Doctor of Philosophy in Mathematics (PhD)","degree_level":"Dissertation","degree_discipline":null,"degree_department":null,"school":null,"contributors":["Arnold, Mark E.","Deaton, Russell J."],"advisors":["Goodman-Strauss, Chaim"],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-08-01T07:00:00Z","date_published":"2011-08-01T07:00:00Z","updated_at":"2026-07-24T00:59:49Z","subjects":["Applied sciences","Pure sciences","Colored squares","Discrete mathematics","Limiting behavior","Nondeterministic fillings","Torus","Applied Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarworks.uark.edu/etd/124","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Arnold, Mark E.","Deaton, Russell J."]},{"key":"dc:contributor.advisor","label":"Advisor","values":["Goodman-Strauss, Chaim"]},{"key":"dc:creator","label":"Author","values":["Rosell Gonzalez, Pablo"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2017-09-29T07:00:00Z"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy in Mathematics (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Applied sciences","Pure sciences","Colored squares","Discrete mathematics","Limiting behavior","Nondeterministic fillings","Torus","Applied Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarworks.uark.edu/etd/124"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Limiting Behavior of Nondeterministic Fillings of the Torus by Colored Squares"]}]}],"canonical_facts":{"dc:contributor":["Arnold, Mark E.","Deaton, Russell J."],"dc:contributor.advisor":["Goodman-Strauss, Chaim"],"dc:creator":["Rosell Gonzalez, Pablo"],"dc:date":["2011"],"dc:date.available":["2017-09-29T07:00:00Z"],"dc:description.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.
"],"dc:identifier":["https://scholarworks.uark.edu/etd/124"],"dc:subject":["Applied sciences","Pure sciences","Colored squares","Discrete mathematics","Limiting behavior","Nondeterministic fillings","Torus","Applied Mathematics"],"dc:title":["Limiting Behavior of Nondeterministic Fillings of the Torus by Colored Squares"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy in Mathematics (PhD)"]},"updated_at":"2026-07-24T00:59:49Z"}