Potts Models on Feynman Diagrams

Abstract

We investigate numerically and analytically Potts models on ``thin'' random graphs -- generic Feynman diagrams, using the idea that such models may be expressed as the N --> 1 limit of a matrix model. The thin random graphs in this limit are locally tree-like, in distinction to the ``fat'' random graphs that appear in the planar Feynman diagram limit, more familiar from discretized models of two dimensional gravity. The interest of the thin graphs is that they give mean field theory behaviour for spin models living on them without infinite range interactions or the boundary problems of genuine tree-like structures such as the Bethe lattice. q-state Potts models display a first order transition in the mean field for q>2, so the thin graph Potts models provide a useful test case for exploring discontinuous transitions in mean field theories in which many quantities can be calculated explicitly in the saddle point approximation.

10 pages, latex, + 6 postscript figures

Cited by in corpus (14)

• Statistical ensemble of scale-free random graphs
• Thin Fisher Zeroes
• Relaxation in graph coloring and satisfiability problems
• Ensemble inequivalence in random graphs
• Frustration effects in antiferromagnets on planar random graphs
• Potts Models with Invisible States on General Bethe Lattices
• Potts Models with (17) Invisible States on Thin Graphs
• The Yang Lee Edge Singularity on Feynman Diagrams
• Vertex Models on Feynman Diagrams
• Yang-Lee Zeros of the Two- and Three-State Potts Model Defined on ${\varphi }^3$ Feynman Diagrams
• Thin Animals
• Finite-size scaling functions of the phase transition in the ferromagnetic Ising model on random regular graphs
• A Potts/Ising Correspondence on Thin Graphs
• Why Loops Don't Matter