Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 087, 7 pages      arXiv:1402.0740

Exact Free Energies of Statistical Systems on Random Networks

Naoki Sasakura a and Yuki Sato b
a) Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
b) National Institute for Theoretical Physics, Department of Physics and Centre for Theoretical Physics, University of the Witwartersrand, WITS 2050, South Africa

Received June 10, 2014, in final form August 07, 2014; Published online August 15, 2014

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.

Key words: random networks; exact results; phase transitions; Ising model; quantum gravity.

pdf (280 kb)   tex (13 kb)


  1. Bachas C., de Calan C., Petropoulos P.M.S., Quenched random graphs, J. Phys. A: Math. Gen. 27 (1994), 6121-6128, hep-th/9405068.
  2. Baxter R.J., Exactly solved models in statistical mechanics, Academic Press, Inc., London, 1982.
  3. Dembo A., Montanari A., Ising models on locally tree-like graphs, Ann. Appl. Probab. 20 (2010), 565-592, arXiv:0804.4726.
  4. Dembo A., Montanari A., Sly A., Sun N., The replica symmetric solution for Potts models on $d$-regular graphs, arXiv:1207.5500.
  5. Dorogovtsev S.N., Goltsev A.V., Mendes J.F.F., Ising model on networks with an arbitrary distribution of connections, Phys. Rev. E 66 (2002), 016104, 5 pages, cond-mat/0203227.
  6. El-Showk S., Paulos M.F., Poland D., Rychkov S., Simmons-Duffin D., Vichi A., Solving the 3D Ising model with the conformal bootstrap, Phys. Rev. D 86 (2012), 025022, 17 pages, arXiv:1203.6064.
  7. Johnston D.A., Plecháč P., Equivalence of ferromagnetic spin models on trees and random graphs, J. Phys. A: Math. Gen. 31 (1998), 475-482.
  8. Kazakov V.A., Ising model on a dynamical planar random lattice: exact solution, Phys. Lett. A 119 (1986), 140-144.
  9. Leone M., Vázquez A., Vespignani A., Zecchina R., Ferromagnetic ordering in graphs with arbitrary degree distribution, Eur. Phys. J. B 28 (2002), 191-197, cond-mat/0203416.
  10. Onsager L., Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. 65 (1944), 117-149.
  11. Sasakura N., Sato Y., Ising model on random networks and the canonical tensor model, Progr. Theoret. Exp. Phys. 2014 (2014), 053B03, 15 pages, arXiv:1401.7806.
  12. Whittle P., Fields and flows on random graphs, in Disorder in Physical Systems, Editors G.R. Grimmett, D. Welsh, Oxford Science Publications, Oxford University Press, New York, 1990, 337-348.

Previous article  Next article   Contents of Volume 10 (2014)