Such discontinuous transitions also appear in multiple Ising models on thin graphs and may have implications for the use of the replica trick in spin-glass models on random graphs. 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. Feynman diagrams have a natural graph representation as seen in Figure 1a, with interaction vertices being represented by nodes, and particles being represented by edges. 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. 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. A computer program which generates all Feynman graphs for given initial and final states and an interaction order in QED (Quantum Electro-Dynamics) is described. non-zero) momenta, so that p(F ) 0 for all two forests, this question has been considered in the fully massive and fully. 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 limit of a matrix model. For general Feynman graphs G, but under the assumption of generic (i.e.
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