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Ising Critical Behavior of Inhomogeneous Curie-Weiss Models and Annealed Random Graphs

Dommers, Sander; Giardinà, Cristian; Giberti, Claudio; van der Hofstad, Remco; Prioriello, Maria Luisa


Sander Dommers

Cristian Giardinà

Claudio Giberti

Remco van der Hofstad

Maria Luisa Prioriello


We study the critical behavior for inhomogeneous versions of the Curie-Weiss model, where the coupling constant Jij(β) for the edge ij on the complete graph is given by Jij(β)=βwiwj/(∑k∈[N]wk) . We call the product form of these couplings the rank-1 inhomogeneous Curie-Weiss model. This model also arises [with inverse temperature β replaced by sinh(β) ] from the annealed Ising model on the generalized random graph. We assume that the vertex weights (wi)i∈[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 τ with τ∈(3,5) , then the critical exponents depend sensitively on τ . In addition, at criticality, the total spin SN satisfies that SN/N(τ−2)/(τ−1) converges in law to some limiting random variable whose distribution we explicitly characterize.


Dommers, S., Giardinà, C., Giberti, C., van der Hofstad, R., & Prioriello, M. L. (2016). Ising Critical Behavior of Inhomogeneous Curie-Weiss Models and Annealed Random Graphs. Communications in mathematical physics, 348(1), 221-263.

Journal Article Type Article
Acceptance Date Jul 13, 2016
Online Publication Date Sep 22, 2016
Publication Date Nov 1, 2016
Deposit Date Jun 29, 2018
Journal Communications in Mathematical Physics
Print ISSN 0010-3616
Electronic ISSN 1432-0916
Publisher Springer Verlag
Peer Reviewed Peer Reviewed
Volume 348
Issue 1
Pages 221-263
Public URL
Publisher URL