Topological, geometric and combinatorial properties of random polyominoes

We study two models of random shapes in this thesis: the Eden Cell Growth Model (EGM) and uniform and percolation distributed polyominoes (also known as lattice-based animals). These models have long been of interest in mathematical physics, probability, and statistical mechanics. Both structures ha...

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Detalles Bibliográficos
Autor: ERIKA BERENICE ROLDAN ROA
Tipo de recurso: tesis doctoral
Estado:Versión aceptada para publicación
Fecha de publicación:2018
País:México
Institución:Centro de Investigación en Matemáticas
Repositorio:Repositorio Institucional CIMAT
OAI Identifier:oai:cimat.repositorioinstitucional.mx:1008/982
Acceso en línea:http://cimat.repositorioinstitucional.mx/jspui/handle/1008/982
Access Level:acceso abierto
Palabra clave:info:eu-repo/classification/MSC/POLINOMIOS
info:eu-repo/classification/cti/1
info:eu-repo/classification/cti/12
info:eu-repo/classification/cti/1208
info:eu-repo/classification/cti/110403
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spelling Topological, geometric and combinatorial properties of random polyominoesERIKA BERENICE ROLDAN ROAinfo:eu-repo/classification/MSC/POLINOMIOSinfo:eu-repo/classification/cti/1info:eu-repo/classification/cti/12info:eu-repo/classification/cti/1208info:eu-repo/classification/cti/110403info:eu-repo/classification/cti/110403We study two models of random shapes in this thesis: the Eden Cell Growth Model (EGM) and uniform and percolation distributed polyominoes (also known as lattice-based animals). These models have long been of interest in mathematical physics, probability, and statistical mechanics. Both structures have interesting topological, combinatorial, and geometrical properties. However, until now, tools from stochastic topology and topological data analysis have not been used to study these properties. By introducing these methods and techniques, we established and proved new results that increase the understanding of topological, geometric, and combinatoric properties of these random models. First, we study the maximum number of holes (the rank of the first homology group) that a polyomino with a given number of tiles can have. We prove a tight bound for the asymptotic behavior and give an exact formula for an infinite sequence of natural numbers for this maximum number of holes. Our second set of contributions of this thesis are about the rate of growth of the expectation of the number of holes in a polyomino with uniform and percolation distributions. We prove the existence of linear bounds for the expected number of holes of a polyomino with respect to both the uniform and percolation distributions. Furthermore, we exhibit particular constants for the upper and lower bounds in the uniform distribution case. Finally, we characterize how the rank of the first homology group, of the stochastic process defined by the EGM, changes in time. This allowed us to design and implement a new algorithm that computes the persistence homology associated to this stochastic process at each time and that keeps track of geometric features of the homology of the process as the area and location of the holes. We present and analyze the results of the computational experiments obtained with this algorithm. We also state conjectures based on these experiments about the asymptotic behavior of the number of holes, the locations of these holes, their associated area, and other geometric and topological properties of this stochastic process.2018-05-10info:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/acceptedVersionapplication/pdfhttp://cimat.repositorioinstitucional.mx/jspui/handle/1008/982reponame:Repositorio Institucional CIMATinstname:Centro de Investigación en Matemáticasinstacron:CIMATinfo:eu-repo/semantics/openAccesshttp://creativecommons.org/licenses/by-nc/4.0oai:cimat.repositorioinstitucional.mx:1008/9822024-08-27T22:38:54Z
dc.title.none.fl_str_mv Topological, geometric and combinatorial properties of random polyominoes
title Topological, geometric and combinatorial properties of random polyominoes
spellingShingle Topological, geometric and combinatorial properties of random polyominoes
ERIKA BERENICE ROLDAN ROA
info:eu-repo/classification/MSC/POLINOMIOS
info:eu-repo/classification/cti/1
info:eu-repo/classification/cti/12
info:eu-repo/classification/cti/1208
info:eu-repo/classification/cti/110403
info:eu-repo/classification/cti/110403
title_short Topological, geometric and combinatorial properties of random polyominoes
title_full Topological, geometric and combinatorial properties of random polyominoes
title_fullStr Topological, geometric and combinatorial properties of random polyominoes
title_full_unstemmed Topological, geometric and combinatorial properties of random polyominoes
title_sort Topological, geometric and combinatorial properties of random polyominoes
dc.creator.none.fl_str_mv ERIKA BERENICE ROLDAN ROA
author ERIKA BERENICE ROLDAN ROA
author_facet ERIKA BERENICE ROLDAN ROA
author_role author
dc.subject.none.fl_str_mv info:eu-repo/classification/MSC/POLINOMIOS
info:eu-repo/classification/cti/1
info:eu-repo/classification/cti/12
info:eu-repo/classification/cti/1208
info:eu-repo/classification/cti/110403
info:eu-repo/classification/cti/110403
topic info:eu-repo/classification/MSC/POLINOMIOS
info:eu-repo/classification/cti/1
info:eu-repo/classification/cti/12
info:eu-repo/classification/cti/1208
info:eu-repo/classification/cti/110403
info:eu-repo/classification/cti/110403
description We study two models of random shapes in this thesis: the Eden Cell Growth Model (EGM) and uniform and percolation distributed polyominoes (also known as lattice-based animals). These models have long been of interest in mathematical physics, probability, and statistical mechanics. Both structures have interesting topological, combinatorial, and geometrical properties. However, until now, tools from stochastic topology and topological data analysis have not been used to study these properties. By introducing these methods and techniques, we established and proved new results that increase the understanding of topological, geometric, and combinatoric properties of these random models. First, we study the maximum number of holes (the rank of the first homology group) that a polyomino with a given number of tiles can have. We prove a tight bound for the asymptotic behavior and give an exact formula for an infinite sequence of natural numbers for this maximum number of holes. Our second set of contributions of this thesis are about the rate of growth of the expectation of the number of holes in a polyomino with uniform and percolation distributions. We prove the existence of linear bounds for the expected number of holes of a polyomino with respect to both the uniform and percolation distributions. Furthermore, we exhibit particular constants for the upper and lower bounds in the uniform distribution case. Finally, we characterize how the rank of the first homology group, of the stochastic process defined by the EGM, changes in time. This allowed us to design and implement a new algorithm that computes the persistence homology associated to this stochastic process at each time and that keeps track of geometric features of the homology of the process as the area and location of the holes. We present and analyze the results of the computational experiments obtained with this algorithm. We also state conjectures based on these experiments about the asymptotic behavior of the number of holes, the locations of these holes, their associated area, and other geometric and topological properties of this stochastic process.
publishDate 2018
dc.date.none.fl_str_mv 2018-05-10
dc.type.none.fl_str_mv info:eu-repo/semantics/doctoralThesis
info:eu-repo/semantics/acceptedVersion
format doctoralThesis
status_str acceptedVersion
dc.identifier.none.fl_str_mv http://cimat.repositorioinstitucional.mx/jspui/handle/1008/982
url http://cimat.repositorioinstitucional.mx/jspui/handle/1008/982
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc/4.0
eu_rights_str_mv openAccess
rights_invalid_str_mv http://creativecommons.org/licenses/by-nc/4.0
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instname:Centro de Investigación en Matemáticas
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instname_str Centro de Investigación en Matemáticas
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institution CIMAT
reponame_str Repositorio Institucional CIMAT
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