Phase transitions in the Ising model on random graphs

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.

Figures (5)

• Figure 1: Bifurcations for Erdős-Rényi (ER) graphon with $p=0.5$ and power-law (PL) graphon with $\alpha=0.2$
• Figure 2: Bifurcation diagrams for \ref{['clim']} for the small-world network with $p=0.05$ and $r=0.1$.
• Figure 3: Eigenvalues of $\boldsymbol{K}$ for small-world graph with $p=0.05$ and $r=0.1$. The first four smallest and largest eigenvalues are plotted in blue and red respectively.
• Figure 4: Spin configurations for: (a) AFM case ($J=-1$) slightly below $T=J\mu_8$, (b) FM case ($J=+1$) slightly below $T=J\mu_0$
• Figure 5: Spin configurations for: (a) FM case slightly below $T=J\mu_1$, (b) FM case slightly below $T=J\mu_2$