On thermodynamic states of the Ising model on scale-free graphs

TL;DR

This paper investigates the thermodynamic states of the Ising model on scale-free graphs with divergent degree moments, showing paramagnetic behavior at high temperatures and nonpercolation in bond percolation models.

Contribution

It introduces a model of scale-free graphs with divergent degree moments and analyzes the thermodynamic phases of the Ising and percolation models on these graphs.

Findings

Ising model is paramagnetic at high temperatures

Bond percolation is nonpercolative for positive probabilities

Graphs are locally similar to uncorrelated complex networks

Abstract

There is proposed a model of scale-free random graphs which are locally close to the uncorrelated complex random networks with divergent $\langle k^2 \rangle$ studied in e.g. S. N. Dorogovtsev {\it et al}, Rev. Mod. Phys. {\bf 80}, 1275 (2008). It is shown that the Ising model on these 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. Possible extensions are discussed.