Ising Model on Networks with an Arbitrary Distribution of Connections

TL;DR

This paper derives the exact critical temperature for the Ising model on random networks with arbitrary degree distributions, revealing anomalous behaviors when the degree distribution has diverging moments.

Contribution

It provides an exact solution for the critical temperature of the Ising model on arbitrary degree distribution networks, highlighting novel phase transition behaviors.

Findings

Critical temperature depends on degree distribution moments.

Anomalous magnetization and susceptibility when degree distribution is fat-tailed.

Infinite critical temperature when second moment diverges.

Abstract

We find the exact critical temperature $T_c$ of the nearest-neighbor ferromagnetic Ising model on an `equilibrium' random graph with an arbitrary degree distribution $P(k)$. We observe an anomalous behavior of the magnetization, magnetic susceptibility and specific heat, when $P(k)$ is fat-tailed, or, loosely speaking, when the fourth moment of the distribution diverges in infinite networks. When the second moment becomes divergent, $T_c$ approaches infinity, the phase transition is of infinite order, and size effect is anomalously strong.