## Edge coloring models and reflection positivity (2007)

Venue: | Journal of the American Mathematical Society |

Citations: | 17 - 0 self |

### BibTeX

@ARTICLE{Szegedy07edgecoloring,

author = {Balázs Szegedy},

title = {Edge coloring models and reflection positivity},

journal = {Journal of the American Mathematical Society},

year = {2007},

volume = {20},

pages = {969--988}

}

### OpenURL

### Abstract

The motivation of this paper comes from statistical physics as well as from combinatorics and topology. The general setup in statistical mechanics can be outlined as follows. Let G be a graph and let C be a finite set of “states ” or “colors”. We think of G as a crystal in which either the edges or the vertices

### Citations

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The Classical Groups, Their Invariants and Representations
- Weyl
- 1946
(Show Context)
Citation Context .... Since pS is invariant in WS = W/W 0 it follows that ¯p is also a preimage of pS under the map W → W/W 0 . Furthermore we have that ¯p is an invariant of Od(R). The first fundamental theorem of Weyl =-=[8]-=- describes the space of invariant elements in W by determining a generating system for it. The elements of this generating system correspond to partitions of the set {1, 2,...,k} into two element subs... |

218 |
Real Algebraic Geometry
- Bochnak, Coste, et al.
- 1998
(Show Context)
Citation Context ...by gS t(v)= ¯ f(xv), v∈ N d , is a real valued edge coloring model which represents the graph parameter f. We will need the following well known consequence of the so-called Positivestellensatz (see: =-=[2]-=-). Theorem 4.11. Let g ∈ R[x1,x2,...,xn] be a polynomial such that it has no root in R n . Then there exist polynomials p,f1,f2,...,fh for some natural number h such that pg =1+f 2 1 + f 2 2 + ···+ f ... |

109 |
The geometry and physics of Knots
- Atiyah
- 1990
(Show Context)
Citation Context ...lgebra and the theory of invariants of the orthogonal group. A topological version of the above described reflection symmetry and reflection positivity arises in topological quantum field theory (see =-=[1]-=- and [3]) where the gluing operation is defined on the formal linear space of manifolds with a fixed boundary. We should also emphasize that the subject has a close connection to pure combinatorics. T... |

96 | Limits of dense graph sequences
- Lovász, Szegedy
(Show Context)
Citation Context ... reflection positive. We show that the Ising model is such an example. Finally we mention that a version of vertex coloring models with an infinite number of states is worked out and characterized in =-=[7]-=-. In such a model the states are elements of a measure space on which the weights are given by a measurable function. From the combinatorial point of view, these vertex coloring models can be regarded... |

58 | Reflection positivity, rank connectivity, and homomorphism of graphs
- Freedman, Lovász, et al.
(Show Context)
Citation Context ...function in edge coloring models is edge reflection positive and is vertex reflection positive in vertex coloring models. A surprising result proved by M. H. Freedman, L. Lovász and A. Schrijver (see =-=[4]-=-) says that vertex reflection positivity is almost enough to characterize the partition functions of vertex coloring models. The extra condition that they need is that the ranks of certain matrices (w... |

17 |
The rank of connection matrices and the dimension of graph algebras
- Lovász
(Show Context)
Citation Context ...ot hard to see [5] that rk(M(k,f)) ≤ dk . Question 3.1. What are the possible sequences rk(M(k,f)) ,k=1, 2, 3,...? The analogy of this question for vertex coloring models was answered by L. Lovász in =-=[6]-=-. The next question is motivated by section 3.1. Question 3.2. Which are those vertex coloring models whose partition functions are edge reflection positive? We know only two examples: The Ising model... |

2 |
Personal communication with
- Freedman, Lovász, et al.
(Show Context)
Citation Context ... times the number of perfect matchings in G. 3.3. Open questions. Let f be an edge coloring model with d colors and let M(k,f) denoteitsk-th connection matrix (see section 2.2). It is not hard to see =-=[5]-=- that rk(M(k,f)) ≤ dk . Question 3.1. What are the possible sequences rk(M(k,f)) ,k=1, 2, 3,...? The analogy of this question for vertex coloring models was answered by L. Lovász in [6]. The next ques... |

1 |
Zhenghan Wang: Universal manifold pairings and positivity http://front.math.ucdavis.edu/math.GT/0503054
- Freedman, Kitaev, et al.
(Show Context)
Citation Context ...nd the theory of invariants of the orthogonal group. A topological version of the above described reflection symmetry and reflection positivity arises in topological quantum field theory (see [1] and =-=[3]-=-) where the gluing operation is defined on the formal linear space of manifolds with a fixed boundary. We should also emphasize that the subject has a close connection to pure combinatorics. The parti... |

1 | Szegedy: Limits of dense graph sequences http://front.math.ucdavis.edu/math.CO/0408173 - Lovász, B |