Exact free energies of statistical systems on random networks

DOI10.3842/SIGMA.2014.087zbMATH Open1297.05226arXiv1402.0740MaRDI QIDQ401283

Publication date: 26 August 2014

Published in: SIGMA. Symmetry, Integrability and Geometry: Methods and Applications (Search for Journal in Brave)

Abstract: 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.

Full work available at URL: https://arxiv.org/abs/1402.0740

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zbMATH Keywords

phase transitions quantum gravity Ising model random networks exact results

Mathematics Subject Classification ID

Small world graphs, complex networks (graph-theoretic aspects) (05C82) Random graphs (graph-theoretic aspects) (05C80) Exactly and quasi-solvable systems arising in quantum theory (81U15) Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics (82B20) Exact solutions to problems in general relativity and gravitational theory (83C15) Dynamical aspects of statistical mechanics (37A60) Applications of functional analysis in statistical physics (46N55)