| 1. |
- Björnberg, Jakob Erik, 1983-
(författare)
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Graphical representations of Ising and Potts models : Stochastic geometry of the quantum Ising model and the space-time Potts model
- 2009
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- HTML clipboard Statistical physics seeks to explain macroscopic properties of matter in terms of microscopic interactions. Of particular interest is the phenomenon of phase transition: the sudden changes in macroscopic properties as external conditions are varied. Two models in particular are of great interest to mathematicians, namely the Ising model of a magnet and the percolation model of a porous solid. These models in turn are part of the unifying framework of the random-cluster representation, a model for random graphs which was first studied by Fortuin and Kasteleyn in the 1970’s. The random-cluster representation has proved extremely useful in proving important facts about the Ising model and similar models. In this work we study the corresponding graphical framework for two related models. The first model is the transverse field quantum Ising model, an extension of the original Ising model which was introduced by Lieb, Schultz and Mattis in the 1960’s. The second model is the space–time percolation process, which is closely related to the contact model for the spread of disease. In Chapter 2 we define the appropriate space–time random-cluster model and explore a range of useful probabilistic techniques for studying it. The space– time Potts model emerges as a natural generalization of the quantum Ising model. The basic properties of the phase transitions in these models are treated in this chapter, such as the fact that there is at most one unbounded fk-cluster, and the resulting lower bound on the critical value in . In Chapter 3 we develop an alternative graphical representation of the quantum Ising model, called the random-parity representation. This representation is based on the random-current representation of the classical Ising model, and allows us to study in much greater detail the phase transition and critical behaviour. A major aim of this chapter is to prove sharpness of the phase transition in the quantum Ising model—a central issue in the theory— and to establish bounds on some critical exponents. We address these issues by using the random-parity representation to establish certain differential inequalities, integration of which gives the results. In Chapter 4 we explore some consequences and possible extensions of the results established in Chapters 2 and 3. For example, we determine the critical point for the quantum Ising model in and in ‘star-like’ geometries.
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| 2. |
- Björner, Anders., et al.
(författare)
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Bier spheres and posets
- 2005
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Ingår i: Discrete & Computational Geometry. - : Springer Science and Business Media LLC. - 0179-5376 .- 1432-0444. ; 34:1, s. 71-86
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Tidskriftsartikel (refereegranskat)abstract
- In 1992 Thomas Bier presented a strikingly simple method to produce a huge number of simplicial (n -2)-spheres on 2n vertices, as deleted joins of a simplicial complex on n vertices with its combinatorial Alexander dual. Here we interpret his construction as giving the poset of all the intervals in a boolean algebra that cut across an ideal. Thus we arrive at a substantial generalization of Bier's construction: the Bier posets Bier(P, I) of an arbitrary bounded poset P of finite length. In the case of face posets of PL spheres this yields cellular generalized Bier spheres. In the case of Eulerian or Cohen-Macaulay posets P we show that the Bier posets Bier(P, I) inherit these properties. In the boolean case originally considered by Bier, we show that all the spheres produced by his construction are shellable, which yields many shellable spheres, most of which lack convex realization. Finally, we present simple explicit formulas for the g-vectors of these simplicial spheres and verify that they satisfy a strong form of the g-conjecture for spheres.
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| 3. |
- Björner, Anders, et al.
(författare)
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On the connectivity of manifold graphs
- 2015
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Ingår i: Proceedings of the American Mathematical Society. - : American Mathematical Society (AMS). - 0002-9939 .- 1088-6826. ; 143:10, s. 4123-4132
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Tidskriftsartikel (refereegranskat)abstract
- This paper is concerned with lower bounds for the connectivity of graphs (one-dimensional skeleta) of triangulations of compact manifolds. We introduce a structural invariant b_M for simplicial d-manifolds M taking values in the range 0 <= b_M <= d-1. The main result is that b_M influences connectivity in the following way: The graph of a d-dimensional simplicial compact manifold M is (2d - b_M)-connected. The parameter b_M has the property that b_M = 0 if the complex M is flag. Hence, our result interpolates between Barnette's theorem (1982) that all d-manifold graphs are (d+1)-connected and Athanasiadis' theorem (2011) that flag d-manifold graphs are 2d-connected. The definition of b_M involves the concept of banner triangulations of manifolds, a generalization of flag triangulations.
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| 4. |
- Jonsson, Jakob, 1972-
(författare)
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Simplicial Complexes of Graphs
- 2005
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Let G be a finite graph with vertex set V and edge set E. A graph complex on G is an abstract simplicial complex consisting of subsets of E. In particular, we may interpret such a complex as a family of subgraphs of G. The subject of this thesis is the topology of graph complexes, the emphasis being placed on homology, homotopy type, connectivity degree, Cohen-Macaulayness, and Euler characteristic. We are particularly interested in the case that G is the complete graph on V. Monotone graph properties are complexes on such a graph satisfying the additional condition that they are invariant under permutations of V. Some well-studied monotone graph properties that we discuss in this thesis are complexes of matchings, forests, bipartite graphs, disconnected graphs, and not 2-connected graphs. We present new results about several other monotone graph properties, including complexes of not 3-connected graphs and graphs not coverable by p vertices. Imagining the vertices as the corners of a regular polygon, we obtain another important class consisting of those graph complexes that are invariant under the natural action of the dihedral group on this polygon. The most famous example is the associahedron, whose faces are graphs without crossings inside the polygon. Restricting to matchings, forests, or bipartite graphs, we obtain other interesting complexes of noncrossing graphs. We also examine a certain "dihedral" variant of connectivity. The third class to be examined is the class of digraph complexes. Some well-studied examples are complexes of acyclic digraphs and not strongly connected digraphs. We present new results about a few other digraph complexes, including complexes of graded digraphs and non-spanning digraphs. Many of our proofs are based on Robin Forman's discrete version of Morse theory. As a byproduct, this thesis provides a loosely defined toolbox for attacking problems in topological combinatorics via discrete Morse theory. In terms of simplicity and power, arguably the most efficient tool is Forman's divide and conquer approach via decision trees, which we successfully apply to a large number of graph and digraph complexes.
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