Ising critical behavior of inhomogeneous Curie-Weiss models and annealed random graphs

Abstract

We study the critical behavior for inhomogeneous versions of the Curie-Weiss model, where the coupling constant $J_{ij}(\beta )$ for the edge $ij$ on the complete graph is given by $J_{ij}(\beta )=\beta w_i{w}_j/({\sum }_{k\in [N]}{w}_k)$. We call the product form of these couplings the rank-1 inhomogeneous Curie-Weiss model. This model also arises (with inverse temperature $\beta$ replaced by $\sinh (\beta )$) from the annealed Ising model on the generalized random graph. We assume that the vertex weights $(w_i{)}_{i\in [N]}$ are regular, in the sense that their empirical distribution converges and the second moment converges as well. We identify the critical temperatures and exponents for these models, as well as a non-classical limit theorem for the total spin at the critical point. These depend sensitively on the number of finite moments of the weight distribution. When the fourth moment of the weight distribution converges, then the critical behavior is the same as on the (homogeneous) Curie-Weiss model, so that the inhomogeneity is weak. When the fourth moment of the weights converges to infinity, and the weights satisfy an asymptotic power law with exponent $\tau$ with $\tau \in (3,5)$, then the critical exponents depend sensitively on $\tau$. In addition, at criticality, the total spin $S_N$ satisfies that $S_N/N^{(\tau -1)/(\tau -2)}$ converges in law to some limiting random variable whose distribution we explicitly characterize.

39 pages. Accepted for publication on Communications in Mathematical Physics

References in corpus (4)

• Critical phenomena in complex networks
• Ising models on power-law random graphs
• Violation of Lee-Yang circle theorem for Ising phase transitions on complex networks
• Quenched central limit theorems for the Ising model on random graphs

Cited by in corpus (6)

• Phase Transitions of the Maximum Likelihood Estimates in the $p$-Spin Curie-Weiss Model
• Nonequilibrium dynamics of the Ising model on heterogeneous networks with an arbitrary distribution of threshold noise
• Large deviations for the annealed Ising model on inhomogeneous random graphs: spins and degrees
• Metastability for Glauber dynamics on the complete graph with coupling disorder
• Berry-Esseen bounds in the inhomogeneous Curie-Weiss model with external field
• Continuous spin models on annealed generalized random graphs