Phase transitions in the Ising model on random graphs

TL;DR

This paper investigates phase transitions in the Ising model on various random graph types using graph limit theory, revealing how critical temperatures relate to eigenvalues and identifying metastable states in small-world networks.

Contribution

It introduces a framework linking phase transitions in the Ising model to graph limits and eigenvalues, with new insights into metastability in small-world graphs.

Findings

Critical temperatures determined by eigenvalues of kernel operators.

Bifurcation diagrams for different graph models.

Metastable behavior observed in small-world networks.

Abstract

We study phase transitions in the Ising model on random graphs using graph limits. We show that the critical temperatures are determined by the eigenvalues of the kernel operator associated with the graph limit. Bifurcation diagrams for Erdos-Renyi, small-world, and power-law graphs illustrate the theory. In the small-world case, we identify metastable behavior in both ferromagnetic and antiferromagnetic regimes.