Vertex models on Feynman diagrams

Abstract

The statistical mechanics of spin models, such as the Ising or Potts models, on generic random graphs can be formulated economically by considering the N → 1 limit of N × N Hermitian matrix models. In this paper we consider the N → 1 limit in complex matrix models, which describes vertex models of different sorts living on random graphs. From the graph theoretic perspective one is using matrix model and field theory inspired methods to count various classes of directed graphs.We also make some remarks on vertex models on planar random graphs (the N → ∞ limit) where the resulting matrix models are not generally soluble using currently known methods. Nonetheless, some particular cases may be mapped onto known models and hence solved.

An iterative solution of the three-colour problem on a random lattice

“…This requires that only configurations where an even number of triangles meet at a given vertex are allowed. Finally, let us mention that the three-colour problem on a random lattice has recently attracted attention as a means of describing vertex models on random graphs [5].…”

Remarks on the eigenvalues distributions of D≤4 Yang–Mills matrix models

“…• The above studied Yang-Mills models with quartic terms are considered in [45], in the context of vertex models on planar graphs, where it is observed that they are equivalent to free fermion models, and the critical behavior is Ising-like since the critical exponents are found to be identical. It is very important to clarify whether these models, or variant thereof, fall indeed in the Ising universality class.…”

“…[κ 1 , ǫ] = dφ dX 1 dX 2 exp 2iκ 1 T rφ[X 1 , X 2 ] − 4ǫ(κ 1 − 2ǫ)T rφ 2 − ǫκ 1 T rX 4ǫ − κ 1 )T rφ 2 . (A 45). …”

Erratum: Rotational symmetry breaking in multimatrix models [Phys. Rev. D66, 085024 (2002)]

“…U (N ) invariance, where N is the size of the matrix) which determines much of the universal behaviour in the large N limit. This global symmetry is present also in all the most relevant multi-matrix models (Ising model on random lattice [6,7], the Q-state Potts model [8,9,10,11,12,13], chain of matrices [14,15,16,17,18,19], models for coloring problem [20,21,22,23,24], vertex models [25,26,27,28,29], the meander model [30,31], the O(n)-model and some generalizations of it [32,33,34,35,36,37,38,39,40,41], and several others [42,43,44,45,46,47,48,49,50]. The list is not complete).…”