Curie-Weiss Model under $\ell^{p}$ constraint and a Generalized Hubbard-Stratonovich Transform
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914933625585664 |
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| author | Dey, Partha S. Kim, Daesung |
| author_facet | Dey, Partha S. Kim, Daesung |
| contents | We consider the Ising Curie-Weiss model on the complete graph constrained under a given $\ell^{p}$ norm for some $p>0$. For $p=\infty$, it reduces to the classical Ising Curie-Weiss model. We prove that for all $p>2$, there exists $β_{c}(p)$ such that for $β<β_{c}(p)$, the magnetization is concentrated at zero and satisfies an appropriate Gaussian CLT. In contrast, for $β>β_{c}(p)$ the magnetization is concentrated at $\pm m_\ast$ for some $m_\ast>0$. We have $β_{c}(p)>1$ for $p>2$ and $\lim_{p\to\infty}β_{c}(p)=3$. We further generalize the model for general symmetric spin distributions and prove a similar phase transition. For $0<p<1$, the log-partition function scales at the order of $n^{2/p-1}$. The proofs are based on a generalized Hubbard-Stratonovich (GHS) transform, which is of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04875 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Curie-Weiss Model under $\ell^{p}$ constraint and a Generalized Hubbard-Stratonovich Transform Dey, Partha S. Kim, Daesung Probability Mathematical Physics 60G50, 60F99, 05C81 (Primary) We consider the Ising Curie-Weiss model on the complete graph constrained under a given $\ell^{p}$ norm for some $p>0$. For $p=\infty$, it reduces to the classical Ising Curie-Weiss model. We prove that for all $p>2$, there exists $β_{c}(p)$ such that for $β<β_{c}(p)$, the magnetization is concentrated at zero and satisfies an appropriate Gaussian CLT. In contrast, for $β>β_{c}(p)$ the magnetization is concentrated at $\pm m_\ast$ for some $m_\ast>0$. We have $β_{c}(p)>1$ for $p>2$ and $\lim_{p\to\infty}β_{c}(p)=3$. We further generalize the model for general symmetric spin distributions and prove a similar phase transition. For $0<p<1$, the log-partition function scales at the order of $n^{2/p-1}$. The proofs are based on a generalized Hubbard-Stratonovich (GHS) transform, which is of independent interest. |
| title | Curie-Weiss Model under $\ell^{p}$ constraint and a Generalized Hubbard-Stratonovich Transform |
| topic | Probability Mathematical Physics 60G50, 60F99, 05C81 (Primary) |
| url | https://arxiv.org/abs/2407.04875 |