Limit Shapes for qVolume Tilings of a Large Hexagon
Lozenges are polygons constructed by gluing two equilateral triangles along an edge. We can fit lozenge pieces together to form larger polygons and given an appropriate polygon we can tile it with lozenges. Lozenge tilings of the semi-regular hexagon with sides A,B,C can be viewed as the 2D picture...
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ftkthstockholm:oai:DiVA.org:kth-280758 2023-05-15T15:18:46+02:00 Limit Shapes for qVolume Tilings of a Large Hexagon Gränsformer i qVolym-plattor för stora hexagon Ahmed, Bako 2020 application/pdf http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-280758 eng eng KTH, Matematik (Avd.) TRITA-SCI-GRU 2020:318 http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-280758 info:eu-repo/semantics/openAccess Combinatorics graph theory orthogonal polynomials Matematik kombinatorik grafteori ortogonala polynom Mathematics Student thesis info:eu-repo/semantics/bachelorThesis text 2020 ftkthstockholm 2022-08-11T12:36:28Z Lozenges are polygons constructed by gluing two equilateral triangles along an edge. We can fit lozenge pieces together to form larger polygons and given an appropriate polygon we can tile it with lozenges. Lozenge tilings of the semi-regular hexagon with sides A,B,C can be viewed as the 2D picture of a stack of cubes in a A x B x C box. In this project we investigate the typical shape of a tiling as the sides A,B,C of the box grow uniformly to infinity and we consider two cases: The uniform case where all tilings occur with equal probability and the q^Volume case where the probability of a tiling is proportional to the volume taken up by the corresponding stack of cubes. To investigate lozenge tilings we transform it into a question on families of non-intersecting paths on a corresponding graph representing the hexagon. Using the Lindström–Gessel–Viennot theorem we can define the probability of a non-intersecting path crossing a particular point in the hexagon both for the uniform and the $q$-Volume case. In each case this probability function is connected to either the Hahn or the $q$-Hahn orthogonal polynomials. The orthogonal polynomials depend on the sides of the hexagon and so we consider the asymptotic behaviour of the polynomials as the sides grow to infinity using a result due to Kuijlaars and Van Assche. This determines the density of non-intersecting paths through every point in the hexagon, which we calculate, and a ``Arctic curve" result which shows that the six corners of the hexagon are (with probability one) tiled with just one type of lozenge. "Lozenger" är polygoner konstruerade genom att limma två liksidiga trianglar längs en kant. Vi kan montera lozengstycken ihop för att bilda större polygoner och med en lämplig polygon kan vi lozengplatta den. Lozengplattor av den semi-liksidiga hexagonen med sidorna A, B, C kan ses som 2D-bilden av en stapel kuber i en A x B x C-box. I det här projektet undersöker vi den typiska formen på en platta när sidorna A, B, C på rutan växer till oändlighet och ... Bachelor Thesis Arctic Royal Institute of Technology, Stockholm: KTHs Publication Database DiVA Arctic |
institution |
Open Polar |
collection |
Royal Institute of Technology, Stockholm: KTHs Publication Database DiVA |
op_collection_id |
ftkthstockholm |
language |
English |
topic |
Combinatorics graph theory orthogonal polynomials Matematik kombinatorik grafteori ortogonala polynom Mathematics |
spellingShingle |
Combinatorics graph theory orthogonal polynomials Matematik kombinatorik grafteori ortogonala polynom Mathematics Ahmed, Bako Limit Shapes for qVolume Tilings of a Large Hexagon |
topic_facet |
Combinatorics graph theory orthogonal polynomials Matematik kombinatorik grafteori ortogonala polynom Mathematics |
description |
Lozenges are polygons constructed by gluing two equilateral triangles along an edge. We can fit lozenge pieces together to form larger polygons and given an appropriate polygon we can tile it with lozenges. Lozenge tilings of the semi-regular hexagon with sides A,B,C can be viewed as the 2D picture of a stack of cubes in a A x B x C box. In this project we investigate the typical shape of a tiling as the sides A,B,C of the box grow uniformly to infinity and we consider two cases: The uniform case where all tilings occur with equal probability and the q^Volume case where the probability of a tiling is proportional to the volume taken up by the corresponding stack of cubes. To investigate lozenge tilings we transform it into a question on families of non-intersecting paths on a corresponding graph representing the hexagon. Using the Lindström–Gessel–Viennot theorem we can define the probability of a non-intersecting path crossing a particular point in the hexagon both for the uniform and the $q$-Volume case. In each case this probability function is connected to either the Hahn or the $q$-Hahn orthogonal polynomials. The orthogonal polynomials depend on the sides of the hexagon and so we consider the asymptotic behaviour of the polynomials as the sides grow to infinity using a result due to Kuijlaars and Van Assche. This determines the density of non-intersecting paths through every point in the hexagon, which we calculate, and a ``Arctic curve" result which shows that the six corners of the hexagon are (with probability one) tiled with just one type of lozenge. "Lozenger" är polygoner konstruerade genom att limma två liksidiga trianglar längs en kant. Vi kan montera lozengstycken ihop för att bilda större polygoner och med en lämplig polygon kan vi lozengplatta den. Lozengplattor av den semi-liksidiga hexagonen med sidorna A, B, C kan ses som 2D-bilden av en stapel kuber i en A x B x C-box. I det här projektet undersöker vi den typiska formen på en platta när sidorna A, B, C på rutan växer till oändlighet och ... |
format |
Bachelor Thesis |
author |
Ahmed, Bako |
author_facet |
Ahmed, Bako |
author_sort |
Ahmed, Bako |
title |
Limit Shapes for qVolume Tilings of a Large Hexagon |
title_short |
Limit Shapes for qVolume Tilings of a Large Hexagon |
title_full |
Limit Shapes for qVolume Tilings of a Large Hexagon |
title_fullStr |
Limit Shapes for qVolume Tilings of a Large Hexagon |
title_full_unstemmed |
Limit Shapes for qVolume Tilings of a Large Hexagon |
title_sort |
limit shapes for qvolume tilings of a large hexagon |
publisher |
KTH, Matematik (Avd.) |
publishDate |
2020 |
url |
http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-280758 |
geographic |
Arctic |
geographic_facet |
Arctic |
genre |
Arctic |
genre_facet |
Arctic |
op_relation |
TRITA-SCI-GRU 2020:318 http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-280758 |
op_rights |
info:eu-repo/semantics/openAccess |
_version_ |
1766348952939003904 |