Combinatorial and probabilistic aspects of lattice path models: (Kombinatorische und probabilistische Aspekte von Gitterwegmodellen)
The first chapter is an introduction which puts the subsequent chapters into the respective scientific contexts. After that, the second chapter deals with random tilings of a hexagon with 60-degree unit rhombi with fixed boundary conditions. Every tiling of a hexagon with integer side lengths r,s,t...
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| Language: | English |
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Niedersächsische Staats- und Universitätsbibliothek
2010
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| Online Access: | http://nbn-resolving.de/urn/resolver.pl?urn=urn:nbn:de:hbz:361-16740 http://deposit.d-nb.de/cgi-bin/dokserv?idn=1003803881 http://bieson.ub.uni-bielefeld.de/volltexte/2010/1674/index.html http://d-nb.info/1003803881/34 http://nbn-resolving.de/urn:nbn:de:hbz:361-16740 |
| Summary: | The first chapter is an introduction which puts the subsequent chapters into the respective scientific contexts. After that, the second chapter deals with random tilings of a hexagon with 60-degree unit rhombi with fixed boundary conditions. Every tiling of a hexagon with integer side lengths r,s,t is considered equally probable and the limiting behaviour is studied as r,s and t tend to infinity proportionally. Certain statistics behave like the eigenvalues of a large random matrix of the Gaussian Unitary Ensemble. This is discussed exhaustively in the literature. In Chapter 2 it is shown that these distributions also occur in the sub-ensemble of symmetric tilings or, equivalently, in tilings of the "half-hexagon". As a by-product an "arctic circle phenomenon" is proven: There is a highly ordered and a highly unordered regime with a sharp boundary between the two being the inscribed ellipse. This phenomenon has so far only been conjectured for this sub-ensemble in a paper by Forrester and Nordenstam. The third chapter is on plane partitions fitting inside an r times s times t box. These are r times s matrices with non-negative integer entries less than or equal to t, such that entries are decreasing monotonically along rows and columns. The volume is the sum of the entries. For fixed r,s,t the uniform distribution is considered and it is shown that the centred and normalised volume random variables converge weakly to the the standard normal distribution as two values of r,s and t tend to infinity. Analogous assertions are shown to hold true for symmetry sub-classes . Univ., Diss.--Bielefeld, 2010 Systemvoraussetzungen: Acrobat reader lzar: ll in der Deutschen Nationalbibliothek gewährleistet |
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