Spin Covariance Fluctuations in the SK Model at High Temperature
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913335762485248 |
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| author | Brennecke, Christian Schertzer, Adrien Van Dam, Chen |
| author_facet | Brennecke, Christian Schertzer, Adrien Van Dam, Chen |
| contents | Based on \cite{H}, it is well known that the rescaled two point correlation functions
\[ \sqrt{N} \langle σ_i ; σ_j\rangle = \sqrt{N} \big( \langle σ_i σ_j\rangle -\langle σ_i\rangle \langle σ_j\rangle\big) \] in the Sherrington-Kirkpatrick spin glass model with non-zero external field admit at sufficiently high temperature an explicit non-Gaussian distributional limit as $N\to \infty$. Inspired by recent results from \cite{ABSY, BSXY, BXY}, we provide a novel proof of the distributional convergence which is based on expanding $\langle σ_i ; σ_j\rangle$ into a sum over suitable weights of self-avoiding paths from vertex $i$ to $j$. Compared to \cite{H}, our key observation is that the path representation of $\langle σ_i ; σ_j\rangle$ provides a direct explanation of the specific form of the limiting distribution of $\sqrt{N} \langle σ_i ; σ_j\rangle$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_19122 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spin Covariance Fluctuations in the SK Model at High Temperature Brennecke, Christian Schertzer, Adrien Van Dam, Chen Probability Mathematical Physics 82D30, 60F05 Based on \cite{H}, it is well known that the rescaled two point correlation functions \[ \sqrt{N} \langle σ_i ; σ_j\rangle = \sqrt{N} \big( \langle σ_i σ_j\rangle -\langle σ_i\rangle \langle σ_j\rangle\big) \] in the Sherrington-Kirkpatrick spin glass model with non-zero external field admit at sufficiently high temperature an explicit non-Gaussian distributional limit as $N\to \infty$. Inspired by recent results from \cite{ABSY, BSXY, BXY}, we provide a novel proof of the distributional convergence which is based on expanding $\langle σ_i ; σ_j\rangle$ into a sum over suitable weights of self-avoiding paths from vertex $i$ to $j$. Compared to \cite{H}, our key observation is that the path representation of $\langle σ_i ; σ_j\rangle$ provides a direct explanation of the specific form of the limiting distribution of $\sqrt{N} \langle σ_i ; σ_j\rangle$. |
| title | Spin Covariance Fluctuations in the SK Model at High Temperature |
| topic | Probability Mathematical Physics 82D30, 60F05 |
| url | https://arxiv.org/abs/2404.19122 |