Pairwise Negative Correlation for Uniform Spanning Subgraphs of the Complete Graph
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910049262108672 |
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| author | Tang, Pengfei Zhang, Zibo |
| author_facet | Tang, Pengfei Zhang, Zibo |
| contents | We investigate the pairwise negative correlation (p-NC) property for uniform probability measures on several families of spanning subgraphs of the complete graph $K_n$. Motivated by conjectured negative dependence properties of the random-cluster model with $q<1$, we focus on three natural families: the set of all connected spanning subgraphs, the set of forests with exactly $k$ components, and the set of connected spanning subgraphs with excess $k$, where $k$ is a fixed integer. We prove that for each of these families, the associated uniform measure satisfies the p-NC property provided $n$ is sufficiently large. Our results extend earlier work on uniform forests and provide the first verification of the p-NC property for uniform connected subgraphs and their truncations on complete graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_10738 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Pairwise Negative Correlation for Uniform Spanning Subgraphs of the Complete Graph Tang, Pengfei Zhang, Zibo Probability Combinatorics We investigate the pairwise negative correlation (p-NC) property for uniform probability measures on several families of spanning subgraphs of the complete graph $K_n$. Motivated by conjectured negative dependence properties of the random-cluster model with $q<1$, we focus on three natural families: the set of all connected spanning subgraphs, the set of forests with exactly $k$ components, and the set of connected spanning subgraphs with excess $k$, where $k$ is a fixed integer. We prove that for each of these families, the associated uniform measure satisfies the p-NC property provided $n$ is sufficiently large. Our results extend earlier work on uniform forests and provide the first verification of the p-NC property for uniform connected subgraphs and their truncations on complete graphs. |
| title | Pairwise Negative Correlation for Uniform Spanning Subgraphs of the Complete Graph |
| topic | Probability Combinatorics |
| url | https://arxiv.org/abs/2603.10738 |