Ising Models on Power-Law Random Graphs

Abstract

We study a ferromagnetic Ising model on random graphs with a power-law degree distribution and compute the thermodynamic limit of the pressure when the mean degree is finite (degree exponent τ > 2), for which the random graph has a tree-like structure. For this, we closely follow the analysis by Dembo and Montanari (Ann. Appl. Probab. 20(2):565-592, 2010) which assumes finite variance degrees (τ > 3), adapting it when necessary and also simplifying it when possible. Our results also apply in cases where the degree distribution does not obey a power law. We further identify the thermodynamic limits of various physical quantities, such as the magnetization and the internal energy. © 2010 The Author(s).