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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2604.20631 |
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| _version_ | 1866916008396062720 |
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| author | Bianchi, Alessandra Jacquier, Vanessa Sfragara, Matteo |
| author_facet | Bianchi, Alessandra Jacquier, Vanessa Sfragara, Matteo |
| contents | We study the Ising model on a two-community stochastic block model, where $n$ spins are split into two equal groups with inter-community interaction parameter $α_n\in[0,1]$. We provide a complete characterization of the phase diagram and show that, almost surely with respect to the graph realization, the model undergoes a uniqueness/non-uniqueness phase transition of the Gibbs measure. In particular, in the supercritical regime, the law of the magnetization vector of the two communities converges to a mixture of Dirac measures that, depending on whether $α_n\gg 1/n$ or $α_n\lesssim1/n$, is supported on two or four points, with possibly different weights. In the uniqueness region, we further analyze the fluctuations of the magnetization vector in the subcritical regime and we prove a quenched central limit theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20631 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Ising Model on a Two-Community Stochastic Block Model Bianchi, Alessandra Jacquier, Vanessa Sfragara, Matteo Probability Statistical Mechanics Mathematical Physics 60J10, 05C80, 05C81 We study the Ising model on a two-community stochastic block model, where $n$ spins are split into two equal groups with inter-community interaction parameter $α_n\in[0,1]$. We provide a complete characterization of the phase diagram and show that, almost surely with respect to the graph realization, the model undergoes a uniqueness/non-uniqueness phase transition of the Gibbs measure. In particular, in the supercritical regime, the law of the magnetization vector of the two communities converges to a mixture of Dirac measures that, depending on whether $α_n\gg 1/n$ or $α_n\lesssim1/n$, is supported on two or four points, with possibly different weights. In the uniqueness region, we further analyze the fluctuations of the magnetization vector in the subcritical regime and we prove a quenched central limit theorem. |
| title | The Ising Model on a Two-Community Stochastic Block Model |
| topic | Probability Statistical Mechanics Mathematical Physics 60J10, 05C80, 05C81 |
| url | https://arxiv.org/abs/2604.20631 |