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

S. Dommers, C. Giardina, C. Giberti, R.W. Hofstad, van der, M.L. Prioriello

### Uittreksel

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 $J_{ij}(\beta)=\beta w_iw_j/(\sum_{k\in[N]}w_k)$. 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.
Originele taal-2 Engels s.n. 38 Gepubliceerd - 2015

### Publicatie series

Naam arXiv 1509.07327 [math.PR]

### Vingerafdruk

Critical Behavior
Random Graphs
Ising
Converge
Moment
Weight Distribution
Critical Exponents
Asymptotic Power
Product Form
Model
Empirical Distribution
Criticality
Critical Temperature
Limit Theorems
Inhomogeneity
Complete Graph
Ising Model
Critical point
Power Law
Limiting

### Citeer dit

Dommers, S., Giardina, C., Giberti, C., Hofstad, van der, R. W., & Prioriello, M. L. (2015). Ising critical behavior of inhomogeneous Curie-Weiss and annealed random graphs. (arXiv; Vol. 1509.07327 [math.PR]). s.n.
Dommers, S. ; Giardina, C. ; Giberti, C. ; Hofstad, van der, R.W. ; Prioriello, M.L. / Ising critical behavior of inhomogeneous Curie-Weiss and annealed random graphs. s.n., 2015. 38 blz. (arXiv).
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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 $J_{ij}(\beta)=\beta w_iw_j/(\sum_{k\in[N]}w_k)$. 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.",
author = "S. Dommers and C. Giardina and C. Giberti and {Hofstad, van der}, R.W. and M.L. Prioriello",
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Dommers, S, Giardina, C, Giberti, C, Hofstad, van der, RW & Prioriello, ML 2015, Ising critical behavior of inhomogeneous Curie-Weiss and annealed random graphs. arXiv, vol. 1509.07327 [math.PR], s.n.

Ising critical behavior of inhomogeneous Curie-Weiss and annealed random graphs. / Dommers, S.; Giardina, C.; Giberti, C.; Hofstad, van der, R.W.; Prioriello, M.L.

s.n., 2015. 38 blz. (arXiv; Vol. 1509.07327 [math.PR]).

TY - BOOK

T1 - Ising critical behavior of inhomogeneous Curie-Weiss and annealed random graphs

AU - Dommers, S.

AU - Giardina, C.

AU - Giberti, C.

AU - Hofstad, van der, R.W.

AU - Prioriello, M.L.

PY - 2015

Y1 - 2015

N2 - 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 $J_{ij}(\beta)=\beta w_iw_j/(\sum_{k\in[N]}w_k)$. 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.

AB - 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 $J_{ij}(\beta)=\beta w_iw_j/(\sum_{k\in[N]}w_k)$. 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.

M3 - Report

T3 - arXiv

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

PB - s.n.

ER -

Dommers S, Giardina C, Giberti C, Hofstad, van der RW, Prioriello ML. Ising critical behavior of inhomogeneous Curie-Weiss and annealed random graphs. s.n., 2015. 38 blz. (arXiv).