### 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, N ? 8, 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. 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.

Original language | English |
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Pages (from-to) | 7349-7363 |

Number of pages | 15 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 30 |

Issue number | 21 |

DOIs | |

Publication status | Published - 7 Nov 1997 |

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## Cite this

*Journal of Physics A: Mathematical and General*,

*30*(21), 7349-7363. https://doi.org/10.1088/0305-4470/30/21/011