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Exact Free Energies of Statistical Systems on Random Networks
Symmetry, integrability and geometry: methods and applications, Tome 10 (2014) Cet article a éte moissonné depuis la source Math-Net.Ru

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Statistical systems on random networks can be formulated in terms of partition functions expressed with integrals by regarding Feynman diagrams as random networks. We consider the cases of random networks with bounded but generic degrees of vertices, and show that the free energies can be exactly evaluated in the thermodynamic limit by the Laplace method, and that the exact expressions can in principle be obtained by solving polynomial equations for mean fields. As demonstrations, we apply our method to the ferromagnetic Ising models on random networks. The free energy of the ferromagnetic Ising model on random networks with trivalent vertices is shown to exactly reproduce that of the ferromagnetic Ising model on the Bethe lattice. We also consider the cases with heterogeneity with mixtures of orders of vertices, and derive the known formula of the Curie temperature.
Keywords: random networks; exact results; phase transitions; Ising model; quantum gravity. 
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     author = {Naoki Sasakura and Yuki Sato},
     title = {Exact {Free} {Energies} of {Statistical} {Systems} on {Random} {Networks}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2014},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a86/}
}
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Naoki Sasakura; Yuki Sato. Exact Free Energies of Statistical Systems on Random Networks. Symmetry, integrability and geometry: methods and applications, Tome 10 (2014). http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a86/

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