Ising models on power-law random graphs

TL;DR

This paper analyzes the thermodynamic properties of ferromagnetic Ising models on power-law random graphs with finite mean degree, extending previous work to include degree distributions with exponent $ au>2$.

Contribution

It adapts and simplifies existing analytical methods to compute the thermodynamic limit of the pressure for graphs with degree exponent $ au>2$, broadening the understanding of Ising models on such networks.

Findings

Derived the thermodynamic limit of pressure for $ au>2$

Identified limits of magnetization and internal energy

Extended analysis to graphs with finite mean degree

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 $\tau>2$), for which the random graph has a tree-like structure. For this, we adapt and simplify an analysis by Dembo and Montanari, which assumes finite variance degrees ($\tau>3$). We further identify the thermodynamic limits of various physical quantities, such as the magnetization and the internal energy.