随机网络上统计系统的精确自由能
Exact Free Energies of Statistical Systems on Random Networks
摘要：
随机网络上的统计系统可以用积分表示的配分函数表示，把费曼图看作随机网络。我们考虑的情况下，随机网络有界的，但通用度的顶点，并表明，自由能可以准确地评估在热力学极限的拉普拉斯方法，并在原则上可以通过求解多项式方程的平均场的精确表达式。作为示范，我们将我们的方法应用于随机网络上的铁磁伊辛模型。铁磁伊辛模型在三价顶点随机网络上的自由能与Bethe晶格上的铁磁伊辛模型的自由能完全相同。我们还考虑了具有混合阶顶点的异质性的情况，并导出了已知的居里温度公式。
英文摘要：
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.
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