Fluctuations for the Sherrington--Kirkpatrick spin glass model near the critical temperature
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914372957241344 |
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| author | Dey, Partha S. Kang, Taegu |
| author_facet | Dey, Partha S. Kang, Taegu |
| contents | We consider the Sherrington--Kirkpatrick spin glass model with zero external field and at inverse temperature $β>0$. Let $F_N(β)$ be the corresponding log-partition function. Under the assumption that $c_N:=N^{1/3}(1-β_N^2)$ is bounded away from $0$, we prove that Var$(F_N(β_N)) = - \frac{1}{2} \log (1-β_N^2) -{β_N^2}/{2} + O( c_N^{-3/2}).$ As a consequence, we obtain Var$(F_N(1-c N^{-1/3})) = \frac16\log N + O(1)$ for any fixed constant $c\in(0,\infty)$. We also prove a Gaussian central limit theorem for the centered and scaled $F_N(β_N)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_05636 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fluctuations for the Sherrington--Kirkpatrick spin glass model near the critical temperature Dey, Partha S. Kang, Taegu Probability 60F05, 82D30 We consider the Sherrington--Kirkpatrick spin glass model with zero external field and at inverse temperature $β>0$. Let $F_N(β)$ be the corresponding log-partition function. Under the assumption that $c_N:=N^{1/3}(1-β_N^2)$ is bounded away from $0$, we prove that Var$(F_N(β_N)) = - \frac{1}{2} \log (1-β_N^2) -{β_N^2}/{2} + O( c_N^{-3/2}).$ As a consequence, we obtain Var$(F_N(1-c N^{-1/3})) = \frac16\log N + O(1)$ for any fixed constant $c\in(0,\infty)$. We also prove a Gaussian central limit theorem for the centered and scaled $F_N(β_N)$. |
| title | Fluctuations for the Sherrington--Kirkpatrick spin glass model near the critical temperature |
| topic | Probability 60F05, 82D30 |
| url | https://arxiv.org/abs/2603.05636 |