A logical approach to concentration
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866910027757912064 |
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| author | Benedikt, Michael Zhukovskii, Maksim |
| author_facet | Benedikt, Michael Zhukovskii, Maksim |
| contents | Concentration results say that a sequence of random variables becomes progressively concentrated around the mean. Such results are common in the study of functions of random graphs. We introduce a real-valued logic with various aggregate operators on graphs, including summation, and prove that every term in the language, seen as a random variable on random graphs within the classical Erdős-Rényi random graph model, is concentrated. We prove this for dense and sparse variants of Erdős-Rényi graphs. On the one hand, our results extend the line of work originating with Fagin and Glebskii et al. on zero-one laws for dense random graphs, as well as the zero-one law of Shelah and Spencer for sparse random graphs. On the other hand, they can be seen as a meta-theorem for inferring concentration results on random graphs, and we give examples of such applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_17752 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A logical approach to concentration Benedikt, Michael Zhukovskii, Maksim Probability Combinatorics Logic Concentration results say that a sequence of random variables becomes progressively concentrated around the mean. Such results are common in the study of functions of random graphs. We introduce a real-valued logic with various aggregate operators on graphs, including summation, and prove that every term in the language, seen as a random variable on random graphs within the classical Erdős-Rényi random graph model, is concentrated. We prove this for dense and sparse variants of Erdős-Rényi graphs. On the one hand, our results extend the line of work originating with Fagin and Glebskii et al. on zero-one laws for dense random graphs, as well as the zero-one law of Shelah and Spencer for sparse random graphs. On the other hand, they can be seen as a meta-theorem for inferring concentration results on random graphs, and we give examples of such applications. |
| title | A logical approach to concentration |
| topic | Probability Combinatorics Logic |
| url | https://arxiv.org/abs/2602.17752 |