Condensed Matter Physics, 2013, vol. 16, No. 2, 23001:1-12

Title: On thermodynamic states of the Ising model on scale-free graphs
  Yu. Kozitsky (Institute of Mathematics, Maria Curie-Sklodowska University, 20-031 Lublin, Poland)

There is proposed a model of scale-free random graphs which are locally close to the uncorrelated complex random networks with divergent < k2> studied in, e.g., S. N. Dorogovtsev et al, Rev. Mod. Phys., 80, 1275 (2008). It is shown that the Ising model on the proposed graphs with interaction intensities of arbitrary signs with probability one is in a paramagnetic state at sufficiently high finite values of the temperature. For the same graphs, the bond percolation model with probability one is in a nonpercolative state for positive values of the percolation probability. These results and their possible extensions are also discussed.

Key words: random graphs, paramagnetic phase, percolation, branching process
PACS: 05.70.Fh, 05.50.+q, 02.50.Ga

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