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Combinatorial Considerations on Two Models from Statistical Mechanics
Linköping University, Department of Computer and Information Science, TCSLAB - Theoretical Computer Science Laboratory. Linköping University, The Institute of Technology.
2007 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

Interactions between combinatorics and statistical mechanics have provided many fruitful insights in both fields. A compelling example is Kuperberg’s solution to the alternating sign matrix conjecture, and its following generalisations. In this thesis we investigate two models from statistical mechanics which have received attention in recent years.

The first is the fully packed loop model. A conjecture from 2001 by Razumov and Stroganov opened the field for a large ongoing investigation of the O(1) loop model and its connections to a refinement of the fully packed loop model. We apply a combinatorial bijection originally found by de Gier to an older conjecture made by Propp.

The second model is the hard particle model. Recent discoveries by Fendley et al. and results by Jonsson suggests that the hard square model with cylindrical boundary conditions possess some beautiful combinatorial properties. We apply both topological and purely combinatorial methods to related independence complexes to try and gain a better understanding of this model.

Place, publisher, year, edition, pages
Matematiska institutionen , 2007. , 7 p.
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1335
Keyword [en]
fully packed loop model, rhombus tilings, hard particle model, independence complex, discrete morse theory
National Category
Discrete Mathematics
URN: urn:nbn:se:liu:diva-10141Local ID: LIU-TEK-LIC-2007:44ISBN: 978-91-85895-44-1OAI: diva2:16889
2007-11-16, 10:15 (English)
Available from: 2007-10-29 Created: 2007-10-29 Last updated: 2010-01-12
List of papers
1. Refined Counting of Fully Packed Loop Configurations
Open this publication in new window or tab >>Refined Counting of Fully Packed Loop Configurations
2007 (English)In: Séminaire Lotharingien de Combinatoire, ISSN 1286-4889, Vol. 56, 1-27 p.Article in journal (Refereed) Published
Abstract [en]

We give a generalisation of a conjecture by Propp on a summation formula for fully packed loop configurations. The original conjecture states that the number of configurations in which each external edge is connected to its neighbour is equal to the total number of configurations of size one less. This conjecture was later generalised by Zuber to include more types of configurations. Our conjecture further refines the counting and provides a general framework for some other summation formulas observed by Zuber. It also implies similar summation formulas for half-turn symmetric configurations.

National Category
urn:nbn:se:liu:diva-12703 (URN)
Available from: 2007-10-29 Created: 2007-10-29
2. Independence Complexes of Cylinders Constructed from Square and Hexagonal Grid Graphs
Open this publication in new window or tab >>Independence Complexes of Cylinders Constructed from Square and Hexagonal Grid Graphs
(English)Manuscript (Other academic)
Abstract [en]

In the paper [Fendley et al., J. Phys. A: Math. Gen., 38 (2005), pp. 315-322], Fendley, Schoutens and van Eerten studied the hard square model at negative activity. They found analytical and numerical evidence that the eigenvalues of the transfer matrix with periodic boundary were all roots of unity. They also conjectured that for an m × n square grid, with doubly periodic boundary, the partition function is equal to 1 when m and n are relatively prime. These conjectures were proven in [Jonsson, Electronic J. Combin., 13(1) (2006), R67]. There, it was also noted that the cylindrical case seemed to have interesting properties when the circumference of the cylinder is odd. In particular, when 3 is a divisor of both the circumference and the width of the cylinder minus 1, the partition function is -2. Otherwise, it is equal to 1. In this paper, we investigate the hard square and hard hexagon models at activity -1, with single periodic boundary, i.e, cylindrical identifications, using both topological and combinatorial techniques. We compute the homology groups of the associated independence complex for small sizes and suggest a matching which, we believe, with further analysis could help solve the conjecture. We also briefly review a technique recently described by Bousquet-M´elou, Linusson and Nevo, for determining some of the eigenvalues of the transfer matrix of the hard square model with cylindrical identification using a related, but more easily analysed model.

National Category
urn:nbn:se:liu:diva-12704 (URN)
Available from: 2007-10-29 Created: 2007-10-29Bibliographically approved

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