Random vectors on the spin configuration of a Curie-Weiss model on Erdos-Renyi random graphs
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929206895575040 |
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| author | Bach, Dominik R. |
| author_facet | Bach, Dominik R. |
| contents | This article is concerned with the asymptotic behaviour of random vectors in a diluted ferromagnetic model. We consider a model introduced by Bovier & Gayrard (1993) with ferromagnetic interactions on a directed Erdős-Rényi random graph. Here, directed connections between graph nodes are uniformly drawn at random with a probability p that depends on the number of nodes N and is allowed to go to zero in the limit. If $Np\longrightarrow\infty$ in this model, Bovier & Gayrard (1993) proved a law of large numbers almost surely, and Kabluchko et al. (2020) proved central limit theorems in probability. Here, we generalise these results for $β<1$ in the regime $Np\longrightarrow\infty$. We show that all those random vectors on the spin configuration that have a limiting distribution under the Curie-Weiss model converge weakly towards the same distribution under the diluted model, in probability on graph realisations. This generalises various results from the Curie-Weiss model to the diluted model. As a special case, we derive a law of large numbers and central limit theorem for two disjoint groups of spins. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2301_04886 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Random vectors on the spin configuration of a Curie-Weiss model on Erdos-Renyi random graphs Bach, Dominik R. Probability Mathematical Physics This article is concerned with the asymptotic behaviour of random vectors in a diluted ferromagnetic model. We consider a model introduced by Bovier & Gayrard (1993) with ferromagnetic interactions on a directed Erdős-Rényi random graph. Here, directed connections between graph nodes are uniformly drawn at random with a probability p that depends on the number of nodes N and is allowed to go to zero in the limit. If $Np\longrightarrow\infty$ in this model, Bovier & Gayrard (1993) proved a law of large numbers almost surely, and Kabluchko et al. (2020) proved central limit theorems in probability. Here, we generalise these results for $β<1$ in the regime $Np\longrightarrow\infty$. We show that all those random vectors on the spin configuration that have a limiting distribution under the Curie-Weiss model converge weakly towards the same distribution under the diluted model, in probability on graph realisations. This generalises various results from the Curie-Weiss model to the diluted model. As a special case, we derive a law of large numbers and central limit theorem for two disjoint groups of spins. |
| title | Random vectors on the spin configuration of a Curie-Weiss model on Erdos-Renyi random graphs |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2301.04886 |