Vertex models on Feynman diagrams |
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Institution: | 1. Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh, EH14 4AS, Scotland, UK;2. Mathematical Institute, 24-29 St Giles'', Oxford, OX1 3LB, UK;1. Graduated Student of Mechanical Engineering, University of Kurdistan, Iran;2. Graduated Student of Aerospace Engineering at K.N Toosi University of Technology, Iran;3. Mechanical Engineering Department, University of Kurdistan, Iran |
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Abstract: | We discuss the statistical mechanics of vertex models on both generic (“thin”) and planar (“fat”) random graphs. Such models can be formulated as the N → 1 and N → ∞ limits of N × N complex matrix models, respectively. From the graph theoretic perspective one is using matrix model and field theory inspired methods to count various classes of directed graphs. For the thin random graphs we use saddle point methods to solve the models in the thermodynamic, large number of vertices limit and note that, as in the case of the eight-vertex model on the square lattice, various other models such as the Ising model appear as particular limits. The generic solution of the fat graph model is rather more elusive, but we show that for several choices of the couplings the models can be reduced to eigenvalue integrals and their critical behaviour deduced. |
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