Concentration via metastable mixing, with applications to the supercritical exponential random graph model
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918264251088896 |
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| author | Winstein, Vilas |
| author_facet | Winstein, Vilas |
| contents | Folklore belief holds that metastable wells in low-temperature statistical mechanics models exhibit high-temperature behavior. We make this rigorous in the exponential random graph model (ERGM) through the lens of concentration of measure. We make use of the supercritical (low-temperature) metastable mixing which was recently proven by Bresler, Nagaraj, and Nichani, and obtain a novel concentration inequality for Lipschitz observables of the ERGM in a large metastable well, answering a question posed by those authors. To achieve this, we prove a new connectivity property for metastable mixing in the ERGM and introduce a new general result yielding concentration inequalities, which extends a result of Chatterjee. We also use a result of Barbour, Brightwell, and Luczak to cover all cases of interest. Our work extends a result of Ganguly and Nam from the subcritical (high-temperature) regime to metastable wells, and we also extend applications of this concentration, namely a central limit theorem for small subcollections of edges and a bound on the Wasserstein distance between the ERGM and the Erdős-Rényi random graph. Finally, to supplement the mathematical content of the article, we present a simulation study of metastable wells in the supercritical ERGM. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_07571 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Concentration via metastable mixing, with applications to the supercritical exponential random graph model Winstein, Vilas Probability Statistical Mechanics Discrete Mathematics Mathematical Physics Statistics Theory 60C05 (Primary) 60B20, 05C80 (Secondary) G.3; G.2.2 Folklore belief holds that metastable wells in low-temperature statistical mechanics models exhibit high-temperature behavior. We make this rigorous in the exponential random graph model (ERGM) through the lens of concentration of measure. We make use of the supercritical (low-temperature) metastable mixing which was recently proven by Bresler, Nagaraj, and Nichani, and obtain a novel concentration inequality for Lipschitz observables of the ERGM in a large metastable well, answering a question posed by those authors. To achieve this, we prove a new connectivity property for metastable mixing in the ERGM and introduce a new general result yielding concentration inequalities, which extends a result of Chatterjee. We also use a result of Barbour, Brightwell, and Luczak to cover all cases of interest. Our work extends a result of Ganguly and Nam from the subcritical (high-temperature) regime to metastable wells, and we also extend applications of this concentration, namely a central limit theorem for small subcollections of edges and a bound on the Wasserstein distance between the ERGM and the Erdős-Rényi random graph. Finally, to supplement the mathematical content of the article, we present a simulation study of metastable wells in the supercritical ERGM. |
| title | Concentration via metastable mixing, with applications to the supercritical exponential random graph model |
| topic | Probability Statistical Mechanics Discrete Mathematics Mathematical Physics Statistics Theory 60C05 (Primary) 60B20, 05C80 (Secondary) G.3; G.2.2 |
| url | https://arxiv.org/abs/2503.07571 |