Order of fluctuations of the free energy in the positive semi-definite MSK model at critical temperature
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915112616460288 |
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| author | Collins-Woodfin, Elizabeth Le, Han Gia |
| author_facet | Collins-Woodfin, Elizabeth Le, Han Gia |
| contents | In this note, we consider the multi-species Sherrington-Kirkpatrick spin glass model at its conjectured critical temperature, and we show that, when the variance profile matrix $Δ^2$ is positive semi-definite, the variance of the free energy is $O(\log^2N)$. Furthermore, when one approaches this temperature threshold from the low temperature side at a rate of $O(N^{-α})$ with $α>0$, the variance is $O(\log^2N+N^{1-α})$. This result is a direct extension of the work of Chen and Lam (2019) who proved an analogous result for the SK model, and our proof methods are adapted from theirs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_11732 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Order of fluctuations of the free energy in the positive semi-definite MSK model at critical temperature Collins-Woodfin, Elizabeth Le, Han Gia Probability Mathematical Physics 60K35, 82B44, 82D30 In this note, we consider the multi-species Sherrington-Kirkpatrick spin glass model at its conjectured critical temperature, and we show that, when the variance profile matrix $Δ^2$ is positive semi-definite, the variance of the free energy is $O(\log^2N)$. Furthermore, when one approaches this temperature threshold from the low temperature side at a rate of $O(N^{-α})$ with $α>0$, the variance is $O(\log^2N+N^{1-α})$. This result is a direct extension of the work of Chen and Lam (2019) who proved an analogous result for the SK model, and our proof methods are adapted from theirs. |
| title | Order of fluctuations of the free energy in the positive semi-definite MSK model at critical temperature |
| topic | Probability Mathematical Physics 60K35, 82B44, 82D30 |
| url | https://arxiv.org/abs/2501.11732 |